From gelbukh at cic.ipn.mx Fri Mar 11 12:39:35 2005
From: gelbukh at cic.ipn.mx (Alexander Gelbukh)
Date: 28/08/02
Subject: FOM: CICLing-2003 -- Computational Linguistics, Mexico, February, Springer LNCS
Message-ID: <200208290434.g7T4Yx718282@pollux.cic.ipn.mx>
CICLing-2003
Fourth International Conference
on Intelligent Text Processing and Computational Linguistics
February 16 to 22, 2003
Mexico City, Mexico
www.CICLing.org
PUBLICATION: Springer LNCS
SUBMISSION: October 10, short papers: October 20
KEYNOTE SPEAKERS:
Eric Brill (Microsoft Research, USA)
Adam Kilgarriff (Brighton U., UK)
Ted Pedersen (U. of Minnesota, USA)
More are likely to be announced, see www.CICLing.org
EXCURSIONS:
Ancient pyramids, Monarch butterflies,
great cave and colonial city, and more.
See photos of past events at www.CICLing.org
URL: http://www.CICLing.org/2003
If you can read our website, please go there and IGNORE the rest
of this document.
+-------------------------------------------------------
| Why CICLing?
+-------------------------------------------------------
CICLing is a small, professional, high-level, very selective,
non-profit conf on Computational Linguistics and Natural Language
Processing.
We consider the following factors to define our success:
GENERAL INTEREST. The conf covers nearly all topics related
to computational linguistics. This makes it attractive for
people from different areas and leads to vivid and interesting
discussions and exchange of opinions.
INFORMAL INTERACTION. It is intended for a small group of
professionals, some 50 participants. This allows for informal
and friendly atmosphere, more resembling a friendly party than
an official event. At CICLing you can pass hours speaking with
your favorite famous scientists who you scarcely could greet in
the crowd at large confs.
EXCELLENT EXCURSIONS. Mexico is a wonderful country rich in
culture, history, and nature. The conference is intended for
people feeling themselves young in their souls, adventurous
explorers in both science and life. Our cultural program brings
the participants to unique marvels of history and nature hidden
from the usual tourists.
RELIEF FROM FROSTS. In the middle of February frosts, the
participants from Europe and North America can enjoy bright warm
sun under the shadow of palms.
+-------------------------------------------------------
| Areas of interest
+-------------------------------------------------------
Areas of interest include, but are not limited by:
Computational linguistics research:
Computational linguistic theories and formalisms
Representation of linguistic knowledge
Morphology
Syntax
Semantics
Discourse models
Text generation
Statistical methods in computational linguistics
Corpus linguistics
Lexical resources
Intelligent text processing and applications:
Document classification and search
Information retrieval
Information extraction
Text mining
Automatic summarization
Spell checking
Natural language interfaces
+-------------------------------------------------------
| Important dates
+-------------------------------------------------------
Submission deadline: October 10, short papers: October 20
Notification of acceptance: November 1, short papers: November 10
Firm camera-ready deadline: November 13
Conf: February 17-23
Authors of rejected full papers will be given a chance to re-submit
their works as short papers before November 5.
+-------------------------------------------------------
| Cultural Program
+-------------------------------------------------------
One of the most exciting things at the conference are excursions to the
ancient Indian pyramids and visiting a unique natural phenomenon, the
Monarch Butterfly wintering site where you can see millions of beautiful
butterflies in the trees and in the air around you. In common opinion of
the last year's participants, the excursions were excellent; at our
webisite you can see their own photos.
Here is the tentative list of excursions:
- The Anthropological Museum: inside Mexico City
- The City Center; tentative and informal
- Teotihuacan: ancient Indian pyramids, 1 hour drive
- Cacahuamilpa and Taxco: great cave and colonial city, 2 hours drive
- Angangueo: Monarch Butterfly wintering site, 4 hours drive
+-------------------------------------------------------
| Program Committee
+-------------------------------------------------------
1. Barbu, Catalina, UK
2. Boitet, Christian, France
3. Bolshakov, Igor, Mexico
4. Bontcheva, Kalina, UK
5. Brusilovsky, Peter, USA
6. Calzolari, Nicoletta, Italy
7. Carroll, John, UK
8. Cassidy, Patrick, USA
9. Cristea, Dan, Romania
10. Gelbukh, Alexander (chair), Mexico
11. Hasida, Koiti, Japan
12. Harada, Yasunari, Japan
13. Hirst, Graeme, Canada
14. Johnson, Frances, UK
15. Kittredge, Richard, USA / Canada
16. Kharrat, Alma, USA
17. Knudsen, Line, Denmark
18. Koch, Gregers, Denmark
19. Kuebler, Sandra, Germany
20. Lappin, Shalom, UK
21. Laufer, Natalia, Russia
22. Lopez-Lopez, Aurelio, Mexico
23. Loukanova, Roussanka, USA / Bulgaria
24. Luedeling, Anke, Germany
25. Maegaard, Bente, Denmark
26. Martin-Vide, Carlos, Spain
27. Mel'cuk, Igor, Canada
28. Metais, Elisabeth, France
29. Mikheev, Andrei, UK
30. Mitkov, Ruslan, UK
31. Murata, Masaki, Japan
32. Narin'yani, Alexander, Russia
33. Nevzorova, Olga, Russia
34. Nirenburg, Sergei, USA
35. Palomar, Manuel, Spain
36. Pedersen, Ted, USA
37. Pineda-Cortes, Luis Alberto, Mexico
38. Piperidis, Stelios, Greece
39. Ren, Fuji, Japan
40. Sag, Ivan, USA
41. Sidorov, Grigori, Mexico
42. Sharoff, Serge, Russia
43. Sun Maosong, China
44. Tait, John, UK
45. Trujillo, Arturo, UK
46. T'sou Ka-yin, Benjamin, Hong Kong
47. Van Guilder, Linda, USA
48. Verspoor, Karin, USA / The Netherlands
49. Vilares Ferro, Manuel, Spain
50. Wilks, Yorick, UK
More info: www.CICLing.org, gelbukh at CICLing.org
Alexander Gelbukh
PC chair
-------------------------------------------------------
I send you this message because I found your address at
a webpage related to the topic of this conf. If you do
not want to receive my messages, please let me know at
gelbukh at CICLing.org. I apologize for inconvenience.
-------------------------------------------------------
From aatu.koskensilta at xortec.fi Tue Mar 1 02:34:56 2005
From: aatu.koskensilta at xortec.fi (Aatu Koskensilta)
Date: Tue, 1 Mar 2005 09:34:56 +0200
Subject: [FOM] Extending the Language of Set Theory
In-Reply-To: <1109527638.24282.79.camel@m66-080-14.mit.edu>
References: <1109527638.24282.79.camel@m66-080-14.mit.edu>
Message-ID: <5de5d5541ed2fc8fa92e4c628f47e7ac@xortec.fi>
On Feb 27, 2005, at 8:07 PM, Dmytro Taranovsky wrote:
> Meaningfully extending the language of set theory opens new horizons
> for
> mathematicians, but any such endeavor also raises a host of
> philosophical issues. I plan to discuss some of the issues in future
> FOM postings.
Lately, I've been playing with ideas closely resembling yours, but from
a slightly different angle. My emphasis has not really been on
addressing the deficiencies of the language of set theory per se, but
rather to see what we can get by trying to push two kinds of reflection
- epistemological and set theoretical - as far as possible. By
epistemological reflection I refer to the kind of reflection formalized
by e.g. various proof theoretical reflection principles, iterated truth
predicates and the like. By set theoretical reflection principles I
refer to principles which are, in some sense, formalizations of the
maxim UNIFY:
Every possible mathematical structure should be exemplified as a set.
There are basically two kinds of attitudes one can adopt in study of
extensions of ZFC by means of such reflection principles. One might be
interested merely to provide an explication of what is "implicit in
acceptance of ZFC" or some particular philosophical and mathematical
position. On the other hand, one might be interested in actually
establishing new mathematical results that are in some sense acceptable
on basis of ZFC and intuitively plausible reasoning. The former leads
to the kind of analysis exemplified in the classical results about
predicative justifiability, the ordinal Gamma_0 and so forth. The
latter is a more risky endeavor, but to me at least more interesting in
the grand scheme of things.
When trying to formally capture UNIFY at least partially one hits the
problem of defining what exactly is a possible mathematical structure.
Any axiomatization of this concept seems to lead to just a new theory
of sets so a more circumspect approach is called for. Luckily, as you
have noted in your paper, various non-set collections - which can be
seen as structures - make sense based on acceptance of ZFC. For
example, the class of all true sentences of the language of set theory
with a constant for every set makes sense, since the concept of set
theoretic truth does. (If we don't accept that there is a determinate
matter of fact as to the truth or falsity of set theoretic sentences
what reason do we have to accept replacement for anything but upwards
absolute formulae?).
Without further ado, let me present the formal system such musings
naturally lead. The language is a two sorted language with a sort for
sets and a sort for classes and a binary predicate for membership. As
axioms for sets we have those of ZFC with replacement and separation as
Pi^1_1 axioms. As to classes we have an axiom saying that V exists, for
every set there corresponds a class of all classes corresponding to the
members of the set and a rule of inference saying that if A is provably
a class ordinal, then for all classes B, the class version of the
constructive hierarchy relative to B up to A exists. This axiom
basically says that if B makes sense, then anything constructible from
it along a well-ordering which makes sense makes sense. As you note,
the resulting theory is interpretable in ZFC+There is an inaccessible
by taking V_kappa as V and the members of L[V_kappa]_alpha as classes,
where alpha is the least ordinal, s.t.
1. alpha > kappa
2. if delta in L[V_kappa]_beta and beta < alpha, then delta < alpha
The advantage of this theory over ZFC+There is an inaccessible is that
"in principle" all of its theorems are acceptable on basis of
acceptance of ZFC. In addition, talk about Skolem functions of V,
elementary substructures of V and so forth can be carried out. And
since classes can be members of other classes, there is no need for
tedious coding trickery.
We come now to set theoretical reflection. Since classes are certainly
possible mathematical structures, there should be, for every class, a
set that is in some sense structurally equivalent to the class. This I
have, tentatively, formalized as follows. We add to the language a new
binary function symbol c which takes a class structure A and a class
ordinal B into a set structure of form with
the following property
for all x in V_kappa( |=phi(x) <=>
|=phi(x))
Again, this is expressed as a rule of inference: if we have established
that B is a class ordinal, then we can infer the above. In addition, we
add the following rule of inference:
Q_1A_1...Q_nA_n( |=phi)
--------------------------------------------------------------
Q_1a_1...Q_na_n( |= phi)
Where Q_i is a sequence of alternating quantifiers and A_i are class
variables and a_i are set variables.
It's a rather trivial exercise to derive various small large cardinal
axioms in this system. However, I don't know how far one can go. There
are several directions for further extensions: the most obvious one is
that the above analysis seems perfectly sensible and acceptable and
hence there should be a natural model of the theory (by UNIFY). The
problem is defining what exactly is a natural model of the theory,
which is muddled by the presence of the special rules of inference.
I'd like to thank you for bringing up this interesting subject. Also,
I'd be interested in the relation of the theory outlined above and the
unfolding of set theory sketched by Solomon Feferman. I'll return to
your paper in more detail as I've had a few moments to digest it all.
--
Aatu Koskensilta (aatu.koskensilta at xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From cnoguera at iiia.csic.es Tue Mar 1 04:57:42 2005
From: cnoguera at iiia.csic.es (mathlog@ub.edu)
Date: Tue, 1 Mar 2005 10:57:42 +0100
Subject: [FOM] ALGEBRAIC AND TOPOLOGICAL METHODS IN NON-CLASSICAL LOGICS II
(Second announcement)
Message-ID:
(apologies for multiple posting)
Second announcement
ALGEBRAIC AND TOPOLOGICAL METHODS IN NON-CLASSICAL LOGICS II
Barcelona, 15-18 June 2005
This meeting shares the goals of the Tbilisi conference with the same
title, held in July 2003, as well as those of the Patras conference on
many-valued logics and residuated structures, held in June 2004.
In recent years the interest in non-classical logics has been growing.
Motivations from computer science, natural language reasoning and
linguistics have played a significant role in this development. The
semantic study of non-classical logics is a field where no single
overarching paradigm has been established, and where a variety of
techniques are currently being explored. An important goal of this
meeting is to promote the cross-fertilization of the fundamental ideas
connected with these approaches. Thus, we aim to bring together
researchers from various fields of non-classical logics and
applications, as well as from lattice theory, universal algebra,
category theory and general topology, in order to foster collaboration
and further research.
The scientific programme of the congress will include a few invited
lectures and will provide ample time for contributed papers and
interaction between participants. Researchers whose interests fit the
general aims of the conference are encouraged to participate. The
featured areas include, but are not limited to, the following (in
alphabetical order):
- Algebraic logic
- Coalgebraic semantics
- Categorical semantics in general
- Dynamic logic and dynamic algebras
- Fuzzy and many-valued logics
- Lattices with operators
- Modal logics
- Ordered topological spaces
- Ordered algebraic structures
- Residuated structures
- Substructural logics
- Topological semantics of modal logic
INVITED SPEAKERS
Guram Bezhanishvili, New Mexico State University, Las Cruces (USA)
Robert Goldblatt, Victoria University, Wellington (New Zealand)
Ian Hodkinson, King's College London (UK)
Peter Jipsen, Chapman University, Orange (USA)
Franco Montagna, Universit? di Siena (Italy)
Hilary Priestley, St. Anne's College, University of Oxford (UK)
James Raftery, University of Natal, Durban (South Africa)
PROGRAMME COMMITTEE
Leo Esakia, Georgian Academy of Sciences
Mai Gehrke, New Mexico State University
Petr H?jek, Academy of Sciences of the Czech Republic
Ramon Jansana, Universitat de Barcelona
Hiroakira Ono, Japan Advanced Institute for Science and Technology
(chair)
Constantine Tsinakis, Vanderbilt University
Yde Venema, Universiteit van Amsterdam
Michael Zacharyaschev, King's College London
ORGANIZING COMMITTEE
Josep Maria Font, Universitat de Barcelona (chair)
?ngel Gil, Universitat Pompeu Fabra (Barcelona)
Jos? Gil, Universitat de Barcelona
Joan Gispert, Universitat de Barcelona
Carles Noguera, Institut d'Investigaci? en Intel?lig?ncia Artificial
(Bellaterra)
Antoni Torrens, Universitat de Barcelona
Ventura Verd?, Universitat de Barcelona
SPONSORING INSTITUTIONS
Ministry of Education and Science (Spanish government)
Department of Universities, Research and Information Society of the
Generalitat de Catalunya (Catalan government)
Faculty of Mathematics of the University of Barcelona
Faculty of Philosophy of the University of Barcelona
Catalan Mathematical Society
With the collaboration of IMUB (Institute of Mathematics, University of
Barcelona) and IIIA (Artificial Intelligence Research Institute, CSIC).
CONTRIBUTED PAPERS
Participants who wish to present a talk should submit an abstract
through the Atlas service (http://atlas-conferences.com/) before 31
March 2005. The abstract should be written in TeX (or in plain,
non-formatted text without formulas) and be at most 2 pages long.
Authors will be notified before 30 April 2005 whether their submission
has been accepted for presentation. Participants needing early
acceptance are advised to submit their abstract as soon as possible and
inform the organizers of the situation.
TRAVEL GRANTS
We hope to provide some funding to partially cover travel expenses of
students and recent Ph.D.'s without grant support, as well as of active
researchers from countries with developing economies. The number and
amount of these grants will depend on the funding available, and will
be paid in cash during the meeting.
Applications should be sent to mathlog at ub.edu before 31 March 2005. To
apply send a message with your personal data, a short CV, a description
of your research area and its relation with the topics of the meeting.
Students and recent Ph.D.'s should also ask their supervisor to send a
letter of support to the same address.
CONGRESS VENUE
The meeting will take place at the Facultat de Matem?tiques of the
Universitat de Barcelona, a 19th century building located in the city
centre. The address is: Gran Via de les Corts Catalanes 585, 08007
Barcelona, Spain. The building is on the square known as Pla?a de la
Universitat (University square). The lecture rooms are on the ground
floor of the building. Participants will be able to use a nearby
computer room with Internet access. No Wi-Fi coverage or free Ethernet
plugs are planned.
REGISTRATION
Participants must register to attend the meeting. The registration fee
is Euro 30 (approx. $ 39 as of March 1st), to be paid upon arrival.
Registration includes conference materials, coffee breaks and snacks,
and a special price for Saturday's dinner. Abstracts of accepted
contributed papers by registered participants will be included in the
congress' booklet.
DEADLINES
Submission of contributed papers: 31 March 2005
Acceptance of contributed papers: 30 April 2005
Travel grant applications: 31 March 2005
MORE INFORMATION
Further details about registration, hotels, schedule, etc., will be
posted at the congress' web page
http://www.mat.ub.edu/~logica/meeting2005/
For more information on the meeting, please visit this page, or write
to mathlog at ub.edu.
?
From aatu.koskensilta at xortec.fi Wed Mar 2 05:44:20 2005
From: aatu.koskensilta at xortec.fi (Aatu Koskensilta)
Date: Wed, 2 Mar 2005 12:44:20 +0200
Subject: [FOM] Extending the Language of Set Theory - addendum
In-Reply-To: <5de5d5541ed2fc8fa92e4c628f47e7ac@xortec.fi>
References: <1109527638.24282.79.camel@m66-080-14.mit.edu>
<5de5d5541ed2fc8fa92e4c628f47e7ac@xortec.fi>
Message-ID:
On Mar 1, 2005, at 9:34 AM, I wrote:
> Again, this is expressed as a rule of inference: if we have established
> that B is a class ordinal, then we can infer the above. In addition, we
> add the following rule of inference:
>
> Q_1A_1...Q_nA_n( |=phi)
> --------------------------------------------------------------
> Q_1a_1...Q_na_n( |= phi)
>
> Where Q_i is a sequence of alternating quantifiers and A_i are class
> variables and a_i are set variables.
I forgot to say anything about why one should accept this rule of
inference! The motivation is this: should "look
like" . In particular, since proper classes are collections
which are not sets and from the "point of view" of V_kappa the
collections that are not in V_kappa are "proper classes" whatever we
can show to hold about proper classes in relation to V should hold
about sets not in V_kappa in relation to V_kappa. The reason we can't
formulate this as an axiom rather than as a rule of inference is that
it's not obvious that the totality of non-set collections is in any
sense determinate - in fact, the opposite is to be expected! Thus we
can't just assume that e.g. "for all classes A, Phi(A)" makes sense at
all. However, we might be able to show that no matter what proper
classes there happen to be - or to put it in another way, which
properties of sets and collections make sense ("are determinate") -
it's impossible that there should be a proper class A, s.t. ~Phi(A),
and thus establish that for all A Phi(A). For example, whatever proper
classes there turns out to be, the intersection of a class and a set is
a set. As an another example whatever proper classes there are, none is
a non-empty subcollection of On u {On} with no epsilon-minimal element
since any such class would show that On is not well-ordered.
It's also noteworthy that the quantifiers Q_1a_1 ... Q_na_n are
unrestricted, i.e. they are not restricted to e.g. the powerset of
V_kappa. This must be so because we have not assumed that non-set
collections are collections of sets. Rather we have assumed that
collections of collections of collections ... are legitimate - it
certainly makes sense to speak of them, and such talk is reducible to
iterated truth predicates. But this means that there is no natural
choice for the restricting set for the quantifiers Q_1 ... Q_n, and
thus we are lead to choose no restriction at all. For example, if
V_kappa is to look like V and for all collections A, Phi(V,A), then
certainly for any *set* a we should have Phi(V_kappa,a).
A possible strenghtening of the rule of inference is as follows:
Ord(C) & Q_1A_1...Q_nA_n(|=phi)
------------------------------------------------------------------------
-------
Q_1A_1...Q_nA_n( A_i in L[V]_C -->
|=phi)
Stretching the similarity of the structure a to the structure A up to
L[V]_C whenever we have proved that C is a (class) ordinal.
--
Aatu Koskensilta (aatu.koskensilta at xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From enayat at american.edu Wed Mar 2 13:50:00 2005
From: enayat at american.edu (Ali Enayat)
Date: Wed, 2 Mar 2005 13:50:00 -0500
Subject: [FOM] Extending Set Theory with Indiscernibles
Message-ID: <004a01c51f58$a58e59c0$f4144d0c@Home>
Taranovsky's interesting recent note [Feb 27, 2005] on extensions of set
theory has inspired me to briefly discuss some recent results that
characterize the rather surprising strength of a natural system of set
theory with a class of indiscernibles, here dubbed ZFCI.
ZFCI is a theory formulated in the language {epsilon, I(x), <}, where I(x)
is a unary predicate [to distinguish the indiscernibles] and < is a binary
relation [for a global well-ordering]. Intuitively speaking, ZFCI is an
extension of ZFC that strongly negates Leibniz's dictum on the identity of
indiscernibles by asserting "there are a proper class of indiscernibles".
The axioms of ZFCI are as follows:
0. The axioms of ZFC;
1. the sentence expressing "I is a proper class of ordinals";
2. a schema expressing that < is a global well-ordering;
3. the Replacement scheme for formulae using I and <; and
4. the Indiscernibility scheme, which is a scheme asserting that (I,<) is a
class of order indiscernibles [in the usual sense of model theory] for
formulae in the language {epsilon, <}.
It turns out that ZFCI goes well beyond ZFC since it proves the existence of
n-Mahlo cardinals for each concrete natural number n. However, if
consistent, it will not prove the statement "for each natural number n,
there is an n-Mahlo cardinal".
Indeed, one can precisely describe the first order consequences of ZFCI in
the usual language of set theory {epsilon}. In order to do so, let PHI be
the set of sentences of following form (where n is a concrete natural
number):
"there is an n-Mahlo cardinal kappa such that V(kappa) is a SIGMA_n
elementary submodel of the universe".
It is not hard to see that ZFC + PHI is equiconsistent with ZFC plus axioms
of the form "there is an n-Mahlo cardinal" (again, where n is a concrete
natural number n). The former theory of course proves the latter theory, but
not vice versa.
Here is the first main result:
Theorem A. For any sentence S in the usual language of set theory {epsilon},
the following two conditions are equivalent:
(i) ZFCI proves S;
(ii) ZFC + PHI proves S.
It is worth pointing out that the situation for Peano arithmetic (or
equivalently: the theory of finite sets) is quite different, i.e., if one
formulates an analogous theory, PAI, extending PA, then the formalized
version of Ramsey's theorem can be used to show that PAI proves precisely
the same arithmetical sentences as PA itself. This adds more plausibility to
the theory ZFCI since it shows that the indiscernibles allow ZFC to
"catch-up" with PA.
One may also wish to *iterate* the idea of adding indiscernibles by adding
countably many new unary predicates I_n (x) for each natural number n in
order to formulate a theory ZFCI# extending ZFCI by adding axioms asserting
that I_(n+1) is a proper class of indiscernibles for formulae in the
language {epsilon, <}augmented with I_1, ., I_n.
As it turns out, this will not buy us any new theorem of set theory that
ZFCI could not prove already, i.e., Theorem A can be improved to the
following result which connects ZFCI and ZFCI! to other systems of set
theory.
Theorem B. For any sentence S in the usual language of set theory {epsilon},
the following five conditions are equivalent:
(i) ZFCI# proves S;
(ii) ZFCI proves S;
(iii) GBC + "the class of ordinals is weakly compact" proves S;
(iv) NFUA proves "S holds in CZ";
(v) ZFC + PHI proves S.
Here GBC is the Godel-Bernays theory of classes; NFUA is the extension of
the Quine-Jensen system of set theory NFU with a universal set obtained by
adding the axioms of Choice, Infinity, and the axiom expressing "every
Cantorian set is strongly Cantorian"; and CZ is the canonical model of ZFC
that can be *interpreted* in models of NFUA.
The equivalence of (i), (ii) and (v) is will be soon available, but the
equivalence of (iii), (iv), and (v) [which were inspired by the work of
Solovay and Holmes] appear in my paper:
Automorphisms, Mahlo Cardinals, and NFU, Nonstandard Models of Arithmetic
and Set Theory, Contemporary Mathematics (Enayat and Kossak, ed.), volume
361, American Mathematical Society, 2004.
Also available on: http://academic2.american.edu/~enayat/Aut.pdf
Best regards,
Ali Enayat
From dmytro at MIT.EDU Wed Mar 2 18:56:05 2005
From: dmytro at MIT.EDU (Dmytro Taranovsky)
Date: Wed, 2 Mar 2005 18:56:05 -0500
Subject: [FOM] Axiomatization through Reflection Principles
Message-ID: <1109807765.42265295677fa@webmail.mit.edu>
A theory of high consistency strength may still be weak at high expression
level. For example, ZFC + "there is a proper class of supercompact cardinals"
(if it is consistent) does not prove existence of Sigma-ZF-4 correct model of
ZFC. Reflection principles can fill in the gap.
A reflection principle states that a certain true theory has a sufficiently
well-behaved model. Well-behaved can mean existing (consistent), Sigma-0-n
correct, well-founded, ... or Sigma-ZF-n correct. The strength of a large
cardinal notion phi can be extended to higher expression levels by writing for
every statement psi, (for every ordinal k (phi(k) --> V(k) satisfies psi))
implies psi. The extension can be weakened by replacing (for every ordinal k
(phi(k) --> V(k) satisfies psi)) with (ZFC proves that for every ordinal k
(phi(k) --> V(k) satisfies psi)). The weaker extension requires lesser
ontological commitment since one does not need to believe phi to accept the
extension.
The kth level of the cumulative hierarchy, V(k), serves as an image of the
universe, so we can formalize vague intuitions about V as requirements (in the
above-mentioned reflection principles) on k, using higher order statements
about V(k) as necessary.
This posting explains some of the topics in my paper
http://web.mit.edu/dmytro/www/NewSetTheory.htm
Dmytro Taranovsky
From codrina.lauth at ais.fraunhofer.de Thu Mar 3 07:54:30 2005
From: codrina.lauth at ais.fraunhofer.de (Ina Lauth)
Date: Thu, 3 Mar 2005 13:54:30 +0100
Subject: [FOM] [Mlnet] WG: [ILP05] MLJ special issue on ILP (2nd Reminder)
Message-ID: <006c01c51ff0$23e41750$e6961a81@laplauth>
****************************************
*** We apologise for multiple copies ***
****************************************
==============
-- SECOND REMINDER --
==============
Machine Learning Journal Special Issue on
INDUCTIVE LOGIC PROGRAMMING
Call for Papers
AIM AND SCOPE:
==============
Following the 14th International Conference on Inductive Logic
Programming held in Porto, Portugal, the Machine Learning Journal
invites authors to submit papers for the Special Issue on Inductive
Logic Programming. In keeping with the original topic of Inductive
Logic Programming, while reflecting also the broadening scope of the
field, authors are invited to submit papers presenting original
results in all aspects of learning logic programs, learning in
first-order logic and in other logic-based knowledge representation
frameworks. Papers on logical aspects of multi-relational learning
and data mining are also welcome. Typical, but not exclusive, topics
of interest for submissions include
* theoretical aspects (logical foundations, computational/statistical
learning theory, specialisation and generalisation operators, etc.)
of learning in logic (logic programs, constraint logic programs,
Datalog, first-order logic, description logics, higher-order logic,
etc.),
* algorithmic aspects of learning in logic including the design of
algorithms along with theoretical and/or empirical analysis,
probabilistic and statistical approaches, distance and kernel-based
methods, relational reinforcement learning, etc.,
* logical foundations of learning and data mining from
multi-relational
databases, scalability issues, inductive databases, link discovery,
* learning and data mining from structured (e.g., labelled graphs,
tree patterns) and semi-structured data (e.g., XML documents),
multi-instance learning,
* efficient implementations and practical applications of
multi-relational learning and data mining in bioinformatics,
computational chemistry, computational linguistics, relational text
and web mining, medical informatics, spatial data mining, etc.
Papers emphasising new topics related to learning in logic, as well as
to
logical foundations of multi-relational learning and data mining are
especially encouraged.
All contributors to ILP 2004 are invited to submit a full paper, but the
special issue is open to everyone. Each submission will be reviewed
according to the standards of the Machine Learning Journal.
IMPORTANT DATES:
================
Titles and short abstracts due: March 15th, 2005
Papers due: April 1st, 2005
Author notification: July 15th, 2005
Final versions of accepted papers due: September 30th, 2005
Publication: Early 2006
GUEST EDITORS:
==============
Ashwin Srinivasan (IBM India Research Laboratory, New Delhi, India)
Rui Camacho (University of Porto, Portugal)
Ross King (University of Wales, Aberystwyth Wales, United
Kingdom)
PAPERS PREVIOUSLY PUBLISHED IN CONFERENCES/WORKSHOPS:
=====================================================
Authors of papers that appeared previously in refereed conferences and
workshops (e.g. in ILP 2004) are encouraged to submit extended versions
of their papers. Such extended papers must be significantly different
from the conference version, as well as accessible to the broad
readership
of the journal, not just to researchers in ILP. More generally, authors
submitting extended versions of previously published conference/workshop
papers are strongly encouraged to expand on the material that was
included
in the original paper, e.g., to provide more details, to give greater,
more
in-depth, discussion of the results and related work, to expand upon the
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of the material (than was possible in a conference paper).
Submissions must not have appeared in, nor be under consideration by,
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previously
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SUBMISSION INFORMATION:
=======================
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with the subject "MLJ special issue on ILP", Include in the body of
the
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Instructions for submission can be found at
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In the instructions to authors that you will find in the above URL do
not forget to read the NOTE on special issues of the journal (you
are required to say in the submitting e-mail that the paper is for a
special issue on ILP). In addition to submitting the paper following
the instructions given in the above URL, please submit it also to
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All enquiries regarding this special issue should be directed to
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I'm looking forward to welcome you to ICML 2005 , 7-11 August 2005 or to
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From tchow at alum.mit.edu Thu Mar 3 13:44:08 2005
From: tchow at alum.mit.edu (Timothy Y. Chow)
Date: Thu, 3 Mar 2005 13:44:08 -0500 (EST)
Subject: [FOM] Complexity of notions/intermediate degrees
Message-ID:
Some weeks ago Joe Shipman wrote:
>Here's a possibly easier question. Does there exist a specific natural
>algorithmic problem whose best known algorithm has an "intermediate"
>complexity? By this I mean that the running time f has the property
>that, for some n>1, the n-fold composition f(f(f(...f(f(x))...) is
>approximately exponential.
The best known algorithms for factoring m have time complexity that is
roughly exp((log m)^e (log log m)^(1-e)) for some e strictly between 0 and
1. I don't think this quite satisfies your definition of "intermediate"
(with x = log m) but morally speaking it seems to be intermediate. But
algorithms for factoring continue to improve.
>The difficulty of finding such a problem is related to the difficulty of
>proving a "gap" between P and NP, I think
Why do you think this? There is no difficulty in proving the time
hierarchy theorem and hence separating P from EXP, even without any
natural examples of intermediates. So it's not clear to me why the
shortage of natural intermediates between P and NP is related to the
difficulty of proving P and NP.
Tim
From JoeShipman at aol.com Thu Mar 3 15:24:54 2005
From: JoeShipman at aol.com (JoeShipman@aol.com)
Date: Thu, 03 Mar 2005 15:24:54 -0500
Subject: [FOM] Complexity of notions/intermediate degrees
Message-ID: <3EE0F85D.6E70B783.0BC565F6@aol.com>
Chow sayeth:
>I don't think this quite satisfies your definition >of "intermediate" (with x = log m) but morally speaking it >seems to be intermediate. ?But algorithms for factoring >continue to improve.
The best algorithms for factoring are not intermediate between polynomial and exponential in my sense. ?Rather, they are functions with the property that for all n and all sufficiently large x,
f(f(...n times...(f(x)))...) < 2^(2^(2^...(n times)))) < f(f(...n+1 times...f(x)...).
This might as well be exponential. ?The "improvements" you speak of, as far as I am concerned, are trivial, except for the single time the exponent e was actually reduced from 1/2 (the original Morrison-Brillhart exponent from 1975) to 1/3 (Number field sieve of Lenstra, Pollard, Pomerance, and others from 1989).
The real issue is that the only natural building blocks we have to construct algorithmic functions from other algorithmic functions involve operators whose complexity is characterized by polynomials, exponentials, and logarithms. ?So we can get a function whose log is a polynomial in the log of x, which isn't much better than polynomial in x, or a function which x is polynomial in the log of, which isn't much worse than exponential.
The reason I think the shortage of natural intermediates is related to the difficulty of separating P from NP is that we don't have a computational operator of intermediate power (to separate P from EXP the strong operator of exponentiation was good enough). ?If there were a natural intermediate function, one could construct a natural operator of intermediate power by using it to bound a search, which could lead to a whole in-between-realm of natural functions which are not in P and not NP-complete. ?(If the true complexity of SAT is exponential, then factoring could be NP-complete but no function of "intermediate" complexity as I define it could be [at least for many-one reducibility, will need to think about the case of Turing reducibility]).
--Joe Shipman
From tchow at alum.mit.edu Thu Mar 3 16:25:05 2005
From: tchow at alum.mit.edu (Timothy Y. Chow)
Date: Thu, 3 Mar 2005 16:25:05 -0500 (EST)
Subject: [FOM] Complexity of notions/intermediate degrees
In-Reply-To: <3EE0F85D.6E70B783.0BC565F6@aol.com>
References: <3EE0F85D.6E70B783.0BC565F6@aol.com>
Message-ID:
On Thu, 3 Mar 2005 JoeShipman at aol.com wrote:
> The reason I think the shortage of natural intermediates is related to
> the difficulty of separating P from NP is that we don't have a
> computational operator of intermediate power (to separate P from EXP the
> strong operator of exponentiation was good enough). ?If there were a
> natural intermediate function, one could construct a natural operator of
> intermediate power by using it to bound a search, which could lead to a
> whole in-between-realm of natural functions which are not in P and not
> NP-complete. ?(If the true complexity of SAT is exponential, then
> factoring could be NP-complete but no function of "intermediate"
> complexity as I define it could be [at least for many-one reducibility,
> will need to think about the case of Turing reducibility]).
I think I follow what you say here, except that I still don't see what
naturalness has to do with anything. We can, after all, construct plenty
of unnatural candidates for things intermediate between P and NP-complete.
Having natural examples would be pleasant, but why would that make proving
that they're computationally inequivalent any easier? There is no
shortage of natural examples of NP-complete problems, but they haven't
been of much use in proving separation theorems.
Also, using the analogy with recursion theory, the absence of natural
intermediate degrees hasn't stopped us from proving that there really are
distinct degrees.
Tim
From drm39 at cam.ac.uk Fri Mar 4 12:10:20 2005
From: drm39 at cam.ac.uk (D.R. MacIver)
Date: 04 Mar 2005 17:10:20 +0000
Subject: [FOM] Hahn Banach and the Baire Property
In-Reply-To:
References: <3EE0F85D.6E70B783.0BC565F6@aol.com>
Message-ID:
Does anyone happen to know (and ideally have a cite for) whether it is
known if the Hahn Banach theorem implies the existence of a subset of R
without the Baire Property (in ZF say)? I'm presuming it does, as it proves
the existence of a non-measurable subset of R, but I can't seem to find a
reference one way or the other.
(Sorry for the rather analysis-like question, but I think it really is
still FOM. Especially as it's entirely possible that the only way to answer
it is to construct a certain model of set theory!)
Thanks,
David R. MacIver
From carlos at science.uva.nl Fri Mar 4 12:56:34 2005
From: carlos at science.uva.nl (Carlos Areces)
Date: Fri, 4 Mar 2005 18:56:34 +0100
Subject: [FOM] Beth Dissertation Price - Last Call for Submissions
Message-ID: <20050304174418.GA23397@remote.science.uva.nl>
E. W. Beth Dissertation Prize: Last Call for Submissions.
Since 2002, FoLLI (the Association of Logic, Language, and Information,
www.folli.org) awards the E. W. Beth Dissertation Prize to outstanding
dissertations in the fields of Logic, Language, and Information. Submissions are
invited for 2005. The prize will be awarded to the best dissertation which
resulted in a Ph.D. in the year 2004. The dissertations will be judged on the
impact they made in their respective fields, breadth and originality of the work,
and also on the interdisciplinarity of the work. Ideally the winning dissertation
will be of interest to researchers in all three fields.
Who qualifies: Those who were awarded a Ph.D. degree in the areas of Logic,
Language, or Information between January 1st, 2004 and December 31st, 2004. There
is no restriction on the nationality of the candidate or the university where the
Ph.D. was granted. However, after a careful consideration, FoLLI has decided to
accept only dissertations written in English.
Prize: The prize consists of
? a certificate
? an invitation to present the thesis during ESSLLI 05
? a donation of 2500 euros provided by the E. W. Beth Foundation.
? fee waive for ESSLLI 05 attendance
? the possibility to publish the thesis (or a revised version of it) in the
new series of books in Logic, Language and Information to be published by
Springer-Verlag as part of LNCS or LNCS/LNAI. (Further information on this
series will be posted on the FoLLI site soon.)
How to submit: We only accept electronic submissions. The following documents are
required:
1. the thesis in pdf or ps format (doc/rtf not accepted);
2. a ten page abstract of the dissertation in ascii or pdf format;
3. a letter of nomination from the thesis supervisor. Self-nominations are not
admitted: each nomination must be sponsored by the thesis supervisor. The
letter of nomination should concisely describe the scope and significance
of the dissertation and state when the degree was officially awarded;
4. two additional letters of support, including at least one letter from a
referee not affiliated with the academic institution that awarded the Ph.D.
degree.
All documents must be submitted electronically to beth_award at dimi.uniud.it . Hard
copy submissions are not admitted.
If you experience any problems with the email submission or do not receive a
notification from us within three working days, please write to
policriti at dimi.uniud.it or folli at inf.unibz.it
* Important dates:
Deadline for Submissions: March 15, 2005.
Notification of Decision: June 30, 2005.
The prize will be officially assigned to the winner at ESSLLI'05
(http://www.macs.hw.ac.uk/esslli05/), the 17th European Summer School in Logic,
Language, and Information, to be held in Edinburgh, Scotland, August 9-19, 2005.
Prize winner will be expected to attend the ceremony and to give a presentation of
her/his Ph.D. dissertation at ESSLLI?05.
Committee
? Anne Abeill? (Universit? Paris 7)
? Johan van Benthem (University of Amsterdam)
? Veronica Dahl (Simon Fraser University)
? Nissim Francez (The Technion, Haifa)
? Valentin Goranko (University of Johannesburg)
? Alessandro Lenci (University of Pisa)
? Ewa Orlowska (Institute of Telecommunications, Poland)
? Gerald Penn (University of Toronto)
? Alberto Policriti (chair) (Universit? di Udine)
? Christian Retor? (Universit? Bordeaux 1 )
? Rob van der Sandt (University of Nijmegen)
? Wolfgang Thomas (RWTH Aachen)
From JoeShipman at aol.com Fri Mar 4 23:27:02 2005
From: JoeShipman at aol.com (JoeShipman@aol.com)
Date: Fri, 04 Mar 2005 23:27:02 -0500
Subject: [FOM] Problem on order types
Message-ID: <30A0609D.6F999AC3.0BC565F6@aol.com>
Here's a cute problem. The answer is a finite integer, but it's very easy to get it wrong.
Let X be an ordered set such that
for all a**
References: <3EE0F85D.6E70B783.0BC565F6@aol.com>
Message-ID:
Sorry, managed to find a reference on my own after all.
The existence of a finitely additive probability measure on P(N) which
vanishes on finite sets is sufficient to give a subset of {0, 1}^N which
lacks the baire property. Specifically (identifying {0, 1}^N with P(N) in
the obvious way) the set { a \in {0, 1}^N : u(a) = 0 } lacks the baire
property.
If you want more details I can either reproduce them here, or you can check
out the reference HAF (page 810, ``Pincus's Pathology'' ).
Thanks,
David MacIver
On Mar 5 2005, D.R. MacIver wrote:
> Does anyone happen to know (and ideally have a cite for) whether it is
> known if the Hahn Banach theorem implies the existence of a subset of R
> without the Baire Property (in ZF say)? I'm presuming it does, as it
> proves the existence of a non-measurable subset of R, but I can't seem to
> find a reference one way or the other.
>
> (Sorry for the rather analysis-like question, but I think it really is
> still FOM. Especially as it's entirely possible that the only way to
> answer it is to construct a certain model of set theory!)
>
> Thanks,
> David R. MacIver
From dmytro at MIT.EDU Sat Mar 5 11:21:15 2005
From: dmytro at MIT.EDU (Dmytro Taranovsky)
Date: Sat, 5 Mar 2005 11:21:15 -0500
Subject: [FOM] On the Strengths of Inaccessible and Mahlo Cardinals
Message-ID: <1110039675.4229dc7b18c2d@webmail.mit.edu>
By using weak set theories, one can clarify the strengths of certain cardinals.
Often, a weak set theory + large cardinal is inter-interpretable (that is there
is a correspondence of models) with a stronger set theory + slightly weaker
cardinals. That large cardinal corresponds to the class
of ordinals of the stronger theory. More precisely,
Theorem:
1. ZFC minus infinity is equiconsistent with rudimentary set theory plus the
axiom of infinity.
2. ZFC minus power set is equiconsistent with rudimentary set theory + there is
an uncountable ordinal.
3. ZFC is equiconsistent with rudimentary set theory + there is an inaccessible
cardinal
4. ZFC + {there is Sigma_n correct inaccessible cardinal}_n is equiconsistent
with rudimentary set theory + there is a Mahlo cardinal.
In general, ZFC + {there is Sigma_n correct large cardinal}_n is
inter-interpretable with rudimentary set theory + there is a regular limit of
stationary many of these cardinals.
Rudimentary set theory is meant to be the weakest set theory in which some basic
things can be done. Levels of Jensen hierarchy for L satisfy it. I am not
sure how strong rudimentary set theory should be--suggestions to that effect
are welcome--but the following version/axiomatization works for the above
theorem: extensionality, foundation, empty set, pairing, union, existence of
transitive closure, existence of the set of all sets with transitive closure
less numerous than a given set, and bounded quantifier separation.
In the theorem, "rudimentary set theory" can be strengthened by extending ZFC
with a stronger "logic", and modifying Sigma_n correct and the replacement
schema to the full expressive power of the logic. Second order logic
corresponds to ZFC minus power set as the weak theory.
More about expressive logics and other topics can be found in my paper:
http://web.mit.edu/dmytro/www/NewSetTheory.htm
Dmytro Taranovsky
From martin at eipye.com Sat Mar 5 12:51:26 2005
From: martin at eipye.com (Martin Davis)
Date: Sat, 05 Mar 2005 09:51:26 -0800
Subject: [FOM] apology from the moderator
Message-ID: <5.1.0.14.2.20050305094415.04264bb0@mail.eipye.com>
Dear FOMers,
These days the number of crude spam items that turn up in the FOM in-box
outnumber legitimate posts. It is easy enough for me to delete these, but
it appears that one item *got away* and was broadcast to you. Unless your
mailer can deal with the Cyrillic alphabet this Russian piece of junk will
have seemed to be pure gibberish.
In any case, I apologize for this lapse.
Martin
From jeremy.clark at wanadoo.fr Sat Mar 5 15:48:21 2005
From: jeremy.clark at wanadoo.fr (Jeremy Clark)
Date: Sat, 5 Mar 2005 21:48:21 +0100
Subject: [FOM] Problem on order types
In-Reply-To: <30A0609D.6F999AC3.0BC565F6@aol.com>
References: <30A0609D.6F999AC3.0BC565F6@aol.com>
Message-ID: <1a28e816f196a8ae62e0e311a49f33e7@wanadoo.fr>
Since we're in FOM, I think it is only fair to be pedantic and say that
the answer is only a finite integer if you assume the law of excluded
middle. To an intuitionist there are many such sets, but I do not know
if a cardinal can be assigned to the class of all such. (Can anyone
enlighten me on this question?) Certainly not a finite one in any case.
If you do assume excluded middle (many don't) then the answer is 11. I
think. (If I am wrong then my excuse is that I don't like to assume
excluded middle. If I am right then please ignore the contents of these
parentheses.)
regards,
Jeremy Clark
On Mar 5, 2005, at 5:27 am, JoeShipman at aol.com wrote:
> Here's a cute problem. The answer is a finite integer, but it's very
> easy to get it wrong.
>
> Let X be an ordered set such that
> for all a****
> How many possibilities are there for the order type of X?
>
> -- JS
>
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom
>
From JoeShipman at aol.com Sat Mar 5 17:36:15 2005
From: JoeShipman at aol.com (JoeShipman@aol.com)
Date: Sat, 05 Mar 2005 17:36:15 -0500
Subject: [FOM] Problem on order types
Message-ID: <762A84A7.7C7A623C.0BC565F6@aol.com>
11 is correct. Congratulations!
From JoeShipman at aol.com Sat Mar 5 20:44:32 2005
From: JoeShipman at aol.com (JoeShipman@aol.com)
Date: Sat, 05 Mar 2005 20:44:32 -0500
Subject: [FOM] Correction to problem on order types
Message-ID: <303CBCAE.5D901400.0BC565F6@aol.com>
In the statement of the problem, I should require that every open interval (a,b) with a****
Message-ID:
Joe,
Can you elaborate what the error is? I got eleven as well. The
empty set; the one point set and then 3 times 3 where the possibilities
are open, closed, or "the long line" for each end.
--Bob Solovay
On Sat, 5 Mar 2005 JoeShipman at aol.com wrote:
> In the statement of the problem, I should require that every open interval (a,b) with a****
> That was the form I had originally come up with the problem, but then I got too clever. Dave Marker claims that changing "reals" to "rationals" allows more solutions than I had contemplated, and pointed out why my comtemplated proof for "rationals" fails.
>
> -- JS
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom
>
From jeremy.clark at wanadoo.fr Sun Mar 6 16:21:07 2005
From: jeremy.clark at wanadoo.fr (Jeremy Clark)
Date: Sun, 6 Mar 2005 22:21:07 +0100
Subject: [FOM] Correction to problem on order types
In-Reply-To:
References:
Message-ID: <123a50f67dfacc2f5bfecddc817f4291@wanadoo.fr>
I don't know if this is the error Joe had in mind, but it did occur to
me after I submitted an answer that there are in fact many more order
types satisfying this condition. Take a set S \subset \omega_1. Then
define A_x to be Q (order type of rationals) if x \in S and 1+Q
otherwise. Let A_S = sum A_x for x \in \omega_1. I *think* one can show
something like: if S \setminus T is stationary in \omega_1 (meets every
closed unbounded set in \omega_1) then A_S cannot be order isomorphic
to A_T. It is possible to define 2^\omega_1 sets such that any two of
them have the property that their setwise difference is stationary. So
it follows that there are 2^\omega_1 sets satisfying Joe's condition.
regards,
Jeremy Clark
On Mar 6, 2005, at 7:05 am, Robert M. Solovay wrote:
> Joe,
>
> Can you elaborate what the error is? I got eleven as well. The
> empty set; the one point set and then 3 times 3 where the possibilities
> are open, closed, or "the long line" for each end.
>
> --Bob Solovay
>
>
>
> On Sat, 5 Mar 2005 JoeShipman at aol.com wrote:
>
>> In the statement of the problem, I should require that every open
>> interval (a,b) with a****> rational numbers.
>>
>> That was the form I had originally come up with the problem, but then
>> I got too clever. Dave Marker claims that changing "reals" to
>> "rationals" allows more solutions than I had contemplated, and
>> pointed out why my comtemplated proof for "rationals" fails.
>>
>> -- JS
>> _______________________________________________
>> FOM mailing list
>> FOM at cs.nyu.edu
>> http://www.cs.nyu.edu/mailman/listinfo/fom
>>
>
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom
>
From marker at math.uic.edu Sun Mar 6 17:35:45 2005
From: marker at math.uic.edu (Dave Marker)
Date: Sun, 6 Mar 2005 16:35:45 -0600 (CST)
Subject: [FOM] joe shipman's question on ordertypes
Message-ID:
Joe Shipman asked how many order types are there for linear orders
where if a****
Theorem: If X is an ordered set such that for all a****
References: <20AFD026.24AD2E18.0BC565F6@aol.com>
Message-ID: <2184.65.24.144.16.1110177090.squirrel@www.math.ohiou.edu>
JoeShipman at aol.com said:
> Can anyone provide an EXPLICIT (choiceless) construction of
> an example not isomorphic to one of the 11 obtained from
> the "real" examples above by replacing the reals with the
> rationals in the obvious way? It's nice to show that there
> are 2^aleph-1 of them with a stationary set argument, but it
> ought to be possible to get just one more without going
> through such advanced combinatorics.
Stick with Dave Marker's solution -- you can't explicitly get
2^aleph-1 distinct-mod-NS sets, but you can get two. If S=omega_1 then
A^S is Q x omega-1; if S=emptyset then A^S is Q + (1+Q) x omega-1.
These are nonisomorphic, as omega-1 embeds continuously into the
latter but not the former.
-----------
Moses Klein
Visiting Assistant Professor
Department of Mathematics
Ohio University
Athens, OH 45701, USA
klein at math.ohiou.edu
From jeremy.clark at wanadoo.fr Mon Mar 7 05:31:22 2005
From: jeremy.clark at wanadoo.fr (Jeremy Clark)
Date: Mon, 7 Mar 2005 11:31:22 +0100
Subject: [FOM] Order types: a proof
In-Reply-To: <20AFD026.24AD2E18.0BC565F6@aol.com>
References: <20AFD026.24AD2E18.0BC565F6@aol.com>
Message-ID:
Joe Shipman wrote:
> Can anyone provide an EXPLICIT (choiceless) construction of
> an example not isomorphic to one of the 11 obtained from
> the "real" examples above by replacing the reals with the
> rationals in the obvious way? ?It's nice to show that there
> are 2^aleph-1 of them with a stationary set argument, but it
> ought to be possible to get just one more without going
> through such advanced combinatorics.
Well, yes: just take \omega_1 copies of (1+Q). That is to say, the
proof for our large collection of stationary sets is still going to
work if we specify a stationary set (in this case \omega_1 itself).
Suppose that this is isomorphic to 1 + (\omega_1 times Q) (the only
contender in your list that it could be isomorphic to). You need
(countable) choice to derive a contradiction, but the construction of
the set itself is choiceless.
Or you can construct (choiceless) Brouwerian counterexamples to your
theorem (trivial ones: subsets of a singleton set, for example, which
cannot be shown to be either empty or non-empty: such a set, trivially
ordered, would be a counter-example to your theorem), but I don't think
that is what you are asking for.
regards,
Jeremy Clark
From jeremy.clark at wanadoo.fr Mon Mar 7 05:42:43 2005
From: jeremy.clark at wanadoo.fr (Jeremy Clark)
Date: Mon, 7 Mar 2005 11:42:43 +0100
Subject: [FOM] Order types: a proof (correction)
In-Reply-To: <20AFD026.24AD2E18.0BC565F6@aol.com>
References: <20AFD026.24AD2E18.0BC565F6@aol.com>
Message-ID:
In my previous e-mail I provided a choiceless construction of a
counter-example, but stated that you need countable choice to prove
non-isomorphism. In fact you don't need choice at all for this proof,
as it can be based on the taking of suprema for countable subsets of
omega_1, which are unique of course.
Regards,
Jeremy Clark
From marker at math.uic.edu Mon Mar 7 08:29:59 2005
From: marker at math.uic.edu (Dave Marker)
Date: Mon, 7 Mar 2005 07:29:59 -0600 (CST)
Subject: [FOM] order types a proof
Message-ID:
Joe asks for an explicit example of two nonisomorphic omega_1-like
orders (say both with bottom element)
Let Q be the rationals and let I= 1+Q or the rationals in [0,1)
Let A be the order \omega_1 x I
Let B be the order 1+\omega_1 x Q (that is \omega_1 x Q with a bottom
element added).
Then A and B are nonisomorphic.
The proof is realatively concrete (but uses \aleph_1 is regular so is
not choiceless--indeed if \aleph_1 is singular they are isomorphic).
Suppose f is an isomorphism. Show that there is an \alpha
such that f maps \alpha x I onto 1+\alpha\times Q
(let \alpha_0 = 0, let \alpha_{n+1} be least such that every element
of \alpha_n x I maps into \alpha_{n+1} x Q and
every element of \alpha_n x Q maps into \alpha_{n+1} x I,
then let \alpha=\sup\alpha_n).
But then we have a contradiction because in A there is a least
element greater that \alpha x I, but in B there is no
least element greater than f(\alpha x I).
Dave
From dmytro at MIT.EDU Mon Mar 7 10:39:20 2005
From: dmytro at MIT.EDU (Dmytro Taranovsky)
Date: Mon, 7 Mar 2005 10:39:20 -0500
Subject: [FOM] Higher Order Set Theory
Message-ID: <1110209960.422c75a87f438@webmail.mit.edu>
Since the universe includes all sets, second order statements about V appear
doubtful or meaningless. However, there is a way to make them meaningful, and
to get a reasonable axiomatization of theory, thus resolving how to deal with
proper classes, "collections" of proper classes, and proper class categories
such as the category of all groups.
The key idea is set theoretical reflection. An ordinal kappa has high
reflection properties by being "large" relative to lower ordinals and by
V(kappa) resembling V. The largeness is best asserted by saying that there are
ordinals below kappa resembling kappa. The universe is limitless, so large
that there are ordinals arbitrarily similar to it with respect to lesser
ordinals.
It is likely that as we reach ordinals of higher and higher reflection
levels, their properties converge. As this happen, ordinary language (first
order logic with membership relation) begins to fail in identifying them: How
does one say that (V(kappa), in) is an elementary substructure of (V, in)? For
any two such ordinals, the theory of (V(kappa), in) is the same.
Ultimately, one reaches the level at which ordinary language cannot
tell the difference in their properties with using a parameter at least as large
as one of the ordinals. At that point, a new word is needed: kappa is a
reflective ordinal, denoted by R(kappa), iff (V, kappa, in) has the same theory
with parameters in V(kappa) as (V, lambda, in) where lambda is any ordinal
>kappa with sufficiently strong reflection properties.
For example, in L every Silver indiscernible has sufficiently strong
reflection properties, so the notion of reflective ordinals for L makes sense.
Reflective ordinals satisfy all large cardinal properties (such as being
inaccessible) that are expressible in ordinary language and are realized in the
universe.
By the reflection principle, in so far as higher order set theory is
meaningful, a higher order statement with parameter x is true about V iff it is
true about V(kappa) with kappa reflective and x in V(kappa). For example, an
infinitary statement phi (can use infinite disjunctions such as x=1 or x=2 or
..., but not infinite strings of quantifiers (except with disjoint scope)) in L
is true about L iff it is true in L_kappa for an indiscernible kappa with phi
in L_kappa. (Full second order logic about L is problematic since
constructibility is not closed under subsets.)
The potential meaning of higher order set theory is thus uniquely
fixed. Therefore, we can officially define higher order semantics (with set
parameters) through reflective ordinals.
I will describe an axiomatization of reflective ordinals in my next posting
(not counting replies). Meanwhile, more information can be found in my paper:
http://web.mit.edu/dmytro/www/NewSetTheory.htm
Dmytro Taranovsky
From enayat at american.edu Mon Mar 7 11:39:40 2005
From: enayat at american.edu (Ali Enayat)
Date: Mon, 7 Mar 2005 11:39:40 -0500
Subject: [FOM] Constructive solutions to the order type puzzle
Message-ID: <002701c52334$43194bd0$c77a4d0c@Home>
This is a partial reply to Joe Shipman's query in connection with the
"order-type" puzzle. In his posting (March 6, 2005), he asked:
"Can anyone provide an EXPLICIT (choiceless) construction of
an example not isomorphic to one of the 11 obtained from
the "real" examples above by replacing the reals with the
rationals in the obvious way? It's nice to show that there
are 2^aleph-1 of them with a stationary set argument, but it
ought to be possible to get just one more without going
through such advanced combinatorics."
It is known that every Borel linear order L is embeddable into a linear
order of the form 2^alpha, where alpha is a countable ordinal, and 2^alpha
is the set of binary sequences of length alpha, endowed with the
lexicographical ordering [Harrington, Marker, Shelah, 1988, Transactions of
American Mathematical Society].
This result implies that every Borel linear order has countable cofinality,
which in turn can be used to prove that in the *Borel realm*, the answer to
the puzzle is reduced to 6 (with the "long line" solutions out of the
picture). Therefore any explicit construction, besides these 6, must be
"highly" nonconstructive.
Regards,
Ali Enayat
From alexzen at com2com.ru Mon Mar 7 17:22:37 2005
From: alexzen at com2com.ru (Alexander Zenkin)
Date: Tue, 8 Mar 2005 01:22:37 +0300
Subject: [FOM] The uncountability of continuum. 1.
In-Reply-To:
Message-ID:
Some FOMers were uttering an opinion that there are two different
proofs of the uncountability of continuum:
1) the direct proof given by Cantor in 1873/4 and
2) the indirect proof (by Reductio ad Absurdum (RAA)) given by
Cantor in 1890/1
(as a special case of the common proof of the inequality |P(Z)| > |Z| for
any set Z).
For example, S.Kleene ("Introduction to metamathematics") gives the
direct proof in the following quite strange (from the classical logic point
of view) form.
THEOREM 1. The set of real numbers, X=(0,1], is uncountable.
(It should be noted that Kleene even does not formulate explicitly this
statement as a THEOREM).
PROOF. Let {A:} "x1, x2, x3, . . . (1) be an infinite list or
enumeration of some but NOT NECESSARY ALL of the real numbers belonging to
the interval X=(0,1]". The application of Cantor's diagonal method (CDM) to
the list (1), generates a new real number, say x*, not belonging to (1).
Consequently, {A:} "the given list (1) is an infinite list or enumeration of
some but NOT ALL of the real numbers belonging to the interval X=(0,1]".
>From the arbitrariness of the list (1), it follows that there is not a
1-1-correspondence between X and N={1,2,3,. . .}, i.e., X is uncountable.
The proof really produces no RAA-contradiction, but has a form of a
quite senseless "deduction" A -- > A.
However it is obvious that Kleene's condition A is in reality a
composition of two opposite cases.
Case 1.
Let {B:} "a list x1, x2, x3, . . . (1) contains NOT ALL real numbers from
X=(0,1]".
The application of CDM to (1), generates a new real number x*, not belonging
to (1). Consequently, {B:} "the given list (1) contains NOT ALL real numbers
from X=(0,1]".
The "deduction" B -- > B is the vicious circle error and therefore
the Case 1 proves nothing from the classical logic point of view. In this
case Cantor's new real number x* is simply a concrete specimen (one of many)
of a real number, not belonging to the list which, according to the
condition B, "contains NOT ALL real numbers from X=(0,1]".
Case 2.
Assume that {C:} "a list x1, x2, x3, . . . (1) contains ALL reals from
X=(0,1]".
The application of CDM to (1), generates a new real number x*, not belonging
to (1). Consequently, {not-C:} "the given list (1) contains NOT ALL reals
from X=(0,1]". From the contradiction (of a very specific form, C -- >
not-C), it follows that C is false. From the arbitrariness of the list (1),
it follows that there is not a 1-1-correspondence between X and N, i.e., X
is uncountable.
In this case Cantor's new real number x* is a counter-example which
disproves a common statement C.
It is obvious that the Case 2 is an indirect proof based on Reductio
ad Absurdum method.
So, the only logical basis in order to state the uncountability of
continuum is Cantor's INDIRECT proof-1890 basing on the Reductio ad Absurdum
method, i.e., the Case 2 above.
We have thus the following important methodological
THESIS-01. There is not a direct proof of the uncountability of continuum.
There is only Cantor's indirect RAA-proof-1890 of the uncountability of
continuum, i.e., the proof by means of Reductio ad Absurdum method.
P.S. Without Thesis-01 I can't explain to my students the fact that some
known set-theorists are deeply wrong stating that there is a direct proof of
the uncountability of continuum. An important philosophical meaning of the
Thesis-01 is obvious, I think.
From rbj01 at rbjones.com Tue Mar 8 03:57:26 2005
From: rbj01 at rbjones.com (Roger Bishop Jones)
Date: Tue, 8 Mar 2005 08:57:26 +0000
Subject: [FOM] Higher Order Set Theory
In-Reply-To: <1110209960.422c75a87f438@webmail.mit.edu>
References: <1110209960.422c75a87f438@webmail.mit.edu>
Message-ID: <05030808572600.01261@localhost.localdomain>
On Monday 07 March 2005 3:39 pm, Dmytro Taranovsky wrote:
> Since the universe includes all sets, second order statements
> about V appear doubtful or meaningless. However, there is a
> way to make them meaningful, and to get a reasonable
> axiomatization of theory, thus resolving how to deal with
> proper classes, "collections" of proper classes, and proper
> class categories such as the category of all groups.
Higher order set theory is a great deal less problematic than
you make it appear.
I have done work in higher order (omega-order) set theory
on and off since 1988 using the theorem provers HOL and
ProofPower, but there are much earlier uses in the literature.
I believe Carnap did some work in higher order set theory
though I can't offer a reference.
As far as metatheory and semantics is concerned, higher order
set theory is hardly more problematic than first order
set theory.
"V", the universe of ALL sets is just as much a problem in
the metatheory of first order set theory as in higher order
set theory.
All the other less controversial models of
first order set theory (e.g. the V(alpha) for alpha inaccessible)
extend to models of w-order set theory (which may or may not be
"standard" models).
As far as axiom systems are concerned, ZFC can straightforwardly
be transcribed into higher order logic, replacing axiom
schemata by axioms quantifying over sets, if you don't mind
(or actually want) something stronger than ZFC.
Francisco Corrella wrote a PhD (back in the eighties)
which addressed higher order set theories which were no
stronger than first order set theory
(though I never understood the motivation for this).
Perhaps you could explain what you think is wrong with
these obvious ways of doing higher order set theory
(apart from losing V as a model).
Roger Jones
From dmytro at MIT.EDU Tue Mar 8 13:49:37 2005
From: dmytro at MIT.EDU (Dmytro Taranovsky)
Date: Tue, 8 Mar 2005 13:49:37 -0500
Subject: [FOM] Axiomatizing Higher Order Set Theory
Message-ID: <1110307777.422df3c11a8c8@webmail.mit.edu>
Some readers were concerned that the argument in my previous posting for
reflective ordinals (informally, ordinals that are "like" the class of
ordinals, Ord) is not sufficiently persuasive. However, the language of set
theory was accepted not because all doubts about set-theoretical truth were
dispelled, but because it filled an epistemological need, had a good philosophy
behind it, and had a reasonable (and consistent) axiomatization.
At the least, axiomatization turns the concept of reflective ordinals into a
valid formal theory. It also makes it easier to debate the existence of
reflective ordinals, and makes reflective ordinals and hence higher order set
theory usable by mathematicians.
Reflective ordinals are regular and hence inaccessible. Existence of Sigma_n
correct inaccessible cardinals for all n makes Ord appear Mahlo, so reflective
ordinals should be Mahlo. One can go on justifying more large cardinal
properties for reflective ordinals.
Reflective ordinals are best axiomatized using elementary embeddings.
Let R be the predicate for reflectiveness. Take an image of the universe and
imagine some sets on top of it. Let j be an elementary embedding of the
extended image with kappa (representing Ord) its critical point. V(kappa) has
the same theory as j(V(kappa)), and kappa is like a reflective ordinal in
j(V(kappa))--more precisely, it is possible that kappa acts as a reflective
ordinal. In that case, we use kappa to define reflectiveness restricted to
kappa: R_j(lambda) iff lambda \in kappa and for all x in V(lambda),
Theory(j(V(kappa)), \in, lambda, x) = Theory(j(V(kappa)), \in, kappa, x). By
elementarity of j, R_j defines reflectiveness for V(kappa). If for every
appropriate (V, kappa, j), phi holds in (V(kappa), \in, R_j), the phi.
The idea can be formalized at the level of indescribable cardinals, but
my formalization is ZFC + For every statement phi, the statement:
If ZFC proves that for every extender with a critical point kappa, (V(k), \in,
R_j) satisfies phi (where j is the corresponding elementary embedding of V into
M), then phi.
Under the assumption that there are no measurable cardinals, the formalization
is relatively complete (at least with respect to large cardinal structure). If
there are measurable cardinals, then the formalization could be strengthened
guided by the fact that reflective cardinals satisfy all genuine large cardinal
properties that are realized in V (and are expressibe in ordinary first order
set theory).
Dmytro Taranovsky
From aberdein at fit.edu Tue Mar 8 20:52:06 2005
From: aberdein at fit.edu (Andrew Aberdein)
Date: Tue, 8 Mar 2005 20:52:06 -0500
Subject: [FOM] Is Gelfond-Schneider constructive?
Message-ID: <8b2cbd5dbcb7538e7940e31b13dbc680@fit.edu>
Can anyone tell me if there is a constructive proof of the
Gelfond-Schneider theorem? (The theorem that if a is algebraic (and
neither 0 or 1) and b is irrational algebraic, then a^b is
transcendental.)
Anne Troelstra implies that there is:
p.9 of turing.wins.uva.nl/~anne/eolss.pdf
I should have been inclined to take his word for it, if it weren't that
Jonathan Borwein says that there isn't:
p. 16 of www.cecm.sfu.ca/personal/jborwein/virtual.pdf
and Pawel Urzyczyn & M.H. Sorensen state that the only proofs they've
seen are non-constructive:
p. 49 of www.mimuw.edu.pl/~urzy/Int/rozdzial2.ps
Can anyone help to resolve my confusion?
Regards, Andrew Aberdein
--
A n d r e w A b e r d e i n, P h. D.
Humanities and Communication,
Florida Institute of Technology,
Melbourne, Florida 32901-6975, U.S.A.
[+1] (321) 674 8368 http://www.fit.edu/~aberdein/
From hendrik at pooq.com Tue Mar 8 23:19:58 2005
From: hendrik at pooq.com (Hendrik Boom)
Date: Tue, 8 Mar 2005 23:19:58 -0500
Subject: [FOM] The uncountability of continuum. 1.
In-Reply-To:
References:
Message-ID: <20050309041958.GA27094@pooq.com>
On Tue, Mar 08, 2005 at 01:22:37AM +0300, Alexander Zenkin wrote:
>
> Some FOMers were uttering an opinion that there are two different
> proofs of the uncountability of continuum:
> 1) the direct proof given by Cantor in 1873/4 and
> 2) the indirect proof (by Reductio ad Absurdum (RAA)) given by
> Cantor in 1890/1
> (as a special case of the common proof of the inequality |P(Z)| > |Z| for
> any set Z).
Let me look at this constructively.
The operative part of any of these proofs is the subproof that, given any
enumeration of real numbers, there is a real number not enumerated. This
can be done constructively.
Whether there is a constructive proof of the uncountability of reals
may depend on just how you define uncountability. If you define it
by saying something like, "NOT (there is a one-to-one correspondence ...)"
uncountability is a negative statement. Now (NOT P) means (P implies FALSE).
A proof of P implies FALSE is a Reductio ad Absurdum (or contains one somewhere
within it).
So constructively, a direct proof of the uncountability of the continuum
is a reduction ad absurdum.
Is there another way to define uncountability? Perhaps by defining it as
"Every countable enumeration leaves something out." If you do that,
there is an obvious direct proof.
-- hendrik
From aatu.koskensilta at xortec.fi Wed Mar 9 06:26:09 2005
From: aatu.koskensilta at xortec.fi (Aatu Koskensilta)
Date: Wed, 9 Mar 2005 13:26:09 +0200
Subject: [FOM] Higher Order Set Theory
In-Reply-To: <05030808572600.01261@localhost.localdomain>
References: <1110209960.422c75a87f438@webmail.mit.edu>
<05030808572600.01261@localhost.localdomain>
Message-ID: <37388b095d12f402eaa5ca055e9d7c8a@xortec.fi>
On Mar 8, 2005, at 10:57 AM, Roger Bishop Jones wrote:
> On Monday 07 March 2005 3:39 pm, Dmytro Taranovsky wrote:
>> Since the universe includes all sets, second order statements
>> about V appear doubtful or meaningless. However, there is a
>> way to make them meaningful, and to get a reasonable
>> axiomatization of theory, thus resolving how to deal with
>> proper classes, "collections" of proper classes, and proper
>> class categories such as the category of all groups.
>
> Higher order set theory is a great deal less problematic than
> you make it appear.
In a sense higher order set theory is not problematic at all and has a
natural axiomatization in Morse-Kelley set theory. However, from a
conceptual point of view, I believe that higher order set theory is
*much* more problematic than the universe of sets, V alone. The problem
is that it's not obvious how there could be a determined totality of
subcollections of V over which the second order quantifiers were to
range over. For if such a determined totality exists, why isn't it just
an another iterative layer on top of those already in V? One could
argue that the idea of proper classes with any substance is against the
idea of set theory as the universal foundational framework for
mathematics, because there is a substantial ontological and conceptual
involvements beyond those reducible to sets.
Also, if the totality of proper classes is a determined totality, there
seems to be no problem in talking about the collection of all proper
classes and so forth. This essentially gives us ZFC + (ZFC for proper
classes) + (ZFC for proper-proper-classes), ..., i.e. we have just a
new hierarchy of collections which looks exactly like the cumulative
hierarchy with a few spots marked (here begin the "proper classes"....
here begin the "proper-proper-classes" ...).
Aatu Koskensilta (aatu.koskensilta at xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From urquhart at cs.toronto.edu Wed Mar 9 09:31:28 2005
From: urquhart at cs.toronto.edu (Alasdair Urquhart)
Date: Wed, 9 Mar 2005 09:31:28 -0500
Subject: [FOM] Is Gelfond-Schneider constructive?
In-Reply-To: <8b2cbd5dbcb7538e7940e31b13dbc680@fit.edu>
References: <8b2cbd5dbcb7538e7940e31b13dbc680@fit.edu>
Message-ID: <200503090931.28991.urquhart@cs.toronto.edu>
I had a look at the version of Gelfond's proof in Hua Loo Keng's
"Introduction to Number Theory." It's less than 3 pages long
and fairly self-contained.
The overall form of the proof is a reductio ad absurdum.
You suppose that alpha, beta and gamma = alpha^beta
are all in some algebraic number field. You then introduce
an integral function R(x) with coefficients determined by a set
of homogeneous linear equations. The proof then proceeds
through a sequence of explicit estimates, using Cauchy's integral
formula, ending with a refutable inequality.
I haven't examined the proof in detail, but the reasoning
appears to be constructive.
From viktormakarov at hotmail.com Wed Mar 9 12:20:05 2005
From: viktormakarov at hotmail.com (Victor Makarov)
Date: Wed, 09 Mar 2005 12:20:05 -0500
Subject: [FOM] Class Abstractions are a Natural Way for Representation of
Mathematical Theories
Message-ID:
Dear FOMers:
Below is an excerpt from my draft paper "Predicat Logic with Classes:
A Natural Approach to Practical Formalization of Mathematics".
The paper (8 pages) is available from the author.
Victor Makarov, vmakarov at acm.org or ViktorMakarov at hotmail.com
---------
It is now a well established view[1, p.215], that any mathematical theory T
is an extension of ZFC set theory by adding to ZFC a finite number of new
constants c_1, ..., c_k and an axiom A(c_1, ..., c_k), (k >= 1),
implicitly defining these constants.
In CL, the theory T can be introduced by using the following theory
definition:
T := {c_1, ..., c_k | A(c_1, ..., c_k)}
That is, mathematical theories in CL are represented as class abstractions
(see an example below).
Any element of the class T, that is, any tuple (z_1, ..., z_k), where z_1,
... , z_k
are some terms of the first-order theory ZFC such that the formula A(z_1,
... , z_k)
is a theorem of ZFC, is called model of the theory T.
Note that this notion of model is different from the notion of model used in
model theory,
it is actually the notion of model [2, p. 190], which has been used in
mathematics well
before the emergence of model theory.
Suppose, that P is a theorem of T and z is a model of T. Let us replace, in
the formula
P, each constant c_i with z_i, (i = 1, ... ,k). As well known[?], the
resulting formula
will be again a theorem (of the host theory, ZFC).
In CL, the corresponding inference rule can be written as follows:
(z \in d, P) => S[d, P, z]
where d is a syntactic variable for class abstractions,
S[d, P, z] is a notation for substitution used in CL.
A theory definition is a special case of the syntactic construct
"definition", which, in
the general case, has the form
T_1 :: M := K
where T_1 is a previously introduced theory, K is a term or a formula
of the theory T_1, M is the name of K. The theory T_1 is the host of M.
If T_1 is the root theory (like ZFC), then T_1 can be omitted.
(The `::'- notation was borrowed from the object-oriented programming
language C++ [3] ).
Note, that K can be again a class abstraction. In that case M is the name of
the theory
represented by the class abstraction K.
The theory definitions are a natural way for formalization of the "little
theories"
approach [4] and the following important advantages(comparing to [4]) can
be mentioned:
1) There is no need for a special syntactic construct for theory
interpetations
(because models can be used instead interpretations);
2) For introducing "little theories" a standard well-known syntactic
construct
is used (class abstractions).
Let us consider the group theory which can be introduced in the following
way:
Group := {G, m| A(G,m)}
where G and m are constants, representing,respectively, the carrier set and
the law of composition and the formula A(G,m) is a conjunction of axioms
of the group theory.
The notion of abelian group could be introduced in the similar way:
AbelGroup := {G, m| A(G,m) \land Comm(m)\};
where Comm(m) is a formula expressing commutativity of the operation m.
But it can be done in a better, structured way:
AbelGroup := Group & Comm(m)
(see a definition of `&' in section~\ref{CL2}).
References
1. J.Dieudonne. A Panorama of Pure Mathematics. New York: Academic Press,
1982.
2. P.Bernays. Axiomatic Set Theory. New York: Dover Publications, 1991.
3. B. Stroustrap. The Design and Evolution of C++. Addison-Wesley, 1994
4. W. M. Farmer, J .D. Guttman and F. J. Thayer: Little Theories. in D.
Kapur, editor,
Automated deduction --- CADE-11, vol. 607 of Lecture Notes in Computer
Science,
pages 567-581. Springer-Verlag, 1992.
From hendrik at pooq.com Wed Mar 9 13:07:53 2005
From: hendrik at pooq.com (Hendrik Boom)
Date: Wed, 9 Mar 2005 13:07:53 -0500
Subject: [FOM] Order types: a proof
In-Reply-To:
References: <20AFD026.24AD2E18.0BC565F6@aol.com>
Message-ID: <20050309180753.GA12462@pooq.com>
On Mon, Mar 07, 2005 at 11:31:22AM +0100, Jeremy Clark wrote:
>
> Or you can construct (choiceless) Brouwerian counterexamples to your
> theorem (trivial ones: subsets of a singleton set, for example, which
> cannot be shown to be either empty or non-empty: such a set, trivially
> ordered, would be a counter-example to your theorem), but I don't think
> that is what you are asking for.
It would be a counterexample if it were possible to show that it not one
of the 11 known solutions. But you can't show it differs from the empty
solution.
-- hendrik
From wwtx at earthlink.net Wed Mar 9 12:50:07 2005
From: wwtx at earthlink.net (William Tait)
Date: Wed, 9 Mar 2005 11:50:07 -0600
Subject: [FOM] The uncountability of continuum. 1.
In-Reply-To: <20050309041958.GA27094@pooq.com>
References:
<20050309041958.GA27094@pooq.com>
Message-ID:
The following, from Hendrik Boom is in response to Alexander Zenkin's
posting about Cantor's two proofs (the 1874 proof by nested intervals
and the 1890 one by the diagonal argument). Zenkin referred to the
former as a direct proof and the latter as indirect.
> So constructively, a direct proof of the uncountability of the
> continuum
> is a reduction ad absurdum.
>
> Is there another way to define uncountability? Perhaps by defining it
> as
> "Every countable enumeration leaves something out." If you do that,
> there is an obvious direct proof.
Yes, there is a constructive proof in Bishop-Bridges of the theorem
that no function from the integers to the reals is onto. But also
Cantor's proof that no function f from a set M to the totality of
subsets of M is onto actually constructs a witness, namely C = {x in M
| x not in f(x)}, and so deserves to be called direct, if not
constructive. The proof that there is no c in M such that C = fc seems
constructive: c in C -> c not in C; and so
c not in C. Hence c in C. So the assumption that C is in the range of f
leads to contradiction, so C witnesses the fact that there is a subset
of M not in the range of f.
It may seem that the argument in the 1874 paper is more direct in that
the witness real number is constructed step-by-step so that, at the nth
step, it is determined not to be any of the first n members in the
given enumeration of reals, whereas the diagonal argument is a
specialization of a general argument that has a bit the look of
logical slight-of-hand. But applied to the case of M = the set of
natural numbers, it can also be viewed as a step-by-step construction
of the witness C such that, at the nth stage, the fact that C is not
one of the first n sets in the given enumeration is already determined
(by having altering the kth diagonal element for k
Since the uncountability of the continuum is a negative statement, a
proof of this statement by contradiction is a constructive proof.
(Proof by contradiction is the standard way to prove a negative
statement constructively!)
Given any enumeration of real numbers (we do not need to make any
assumption about whether it contains all real numbers or not: this is
the mistake in Zenkin's exposition) we produce a real number which is
not in the range of that enumeration by an explicit computation (some
variant of Cantor's diagonal method). So we have a procedure which,
given any constructive proof that the continuum can be enumerated (a
specific enumeration of reals and a constructive proof that any real
belongs to its range), would generate a contradiction. That is what
it means to prove "the continuum cannot be enumerated" constructively.
Sincerely, Randall Holmes
From dmytro at MIT.EDU Wed Mar 9 16:25:41 2005
From: dmytro at MIT.EDU (Dmytro Taranovsky)
Date: Wed, 9 Mar 2005 16:25:41 -0500
Subject: [FOM] Higher Order Set Theory
In-Reply-To: <05030808572600.01261@localhost.localdomain>
References: <1110209960.422c75a87f438@webmail.mit.edu>
<05030808572600.01261@localhost.localdomain>
Message-ID: <1110403541.422f69d53859e@webmail.mit.edu>
Roger Bishop Jones wrote:
>Higher order set theory is a great deal less problematic than
>you make it appear.
The problem is whether higher order statements have any meaning or
are just symbols on paper. It is possible to axiomatize something
that looks like higher order set theory, and severals proposals
have been made. However, without semantics or guiding ideas, we
have no way to choose the formalization. For example, we would not
know whether to include the axiom of global choice.
Second order logic about, say, integers is meaningful because a
predicate on integers can be treated/defined as a set of integers.
By contrast, the universe includes every set, that is every
collection of objects.
If all predicates on sets had their own independent existence, then
we could make sets of proper class predicates, contradicting the
totality of the universe. Since they do not, we have to explain
what does it mean that there is a predicate satisfying such and
such conditions. (We can still talk about particular predicates.)
Third order set theory is still more problematic.
Fortunately, we do not have to debate metaphysical meaningfulness
of the notion of existence of a property. By using reflective
ordinals, we can recharacterize questions about predicates as
questions about sets.
Also, for ordinary set theory, we do not have to claim that V
exists. We could say that the universe is a convenient figure of
speech, and, for example, translate "large cardinals properties
realized in V" as "large cardinal properties for which there is a
set satisfying the property".
Dmytro Taranovsky
http://web.mit.edu/dmytro/www/main.htm
From jeremy.clark at wanadoo.fr Wed Mar 9 17:50:50 2005
From: jeremy.clark at wanadoo.fr (Jeremy Clark)
Date: Wed, 9 Mar 2005 23:50:50 +0100
Subject: [FOM] Order types: a proof
In-Reply-To: <20050309180753.GA12462@pooq.com>
References: <20AFD026.24AD2E18.0BC565F6@aol.com>
<20050309180753.GA12462@pooq.com>
Message-ID:
On Mar 9, 2005, at 7:07 pm, Hendrik Boom wrote:
> On Mon, Mar 07, 2005 at 11:31:22AM +0100, Jeremy Clark wrote:
>>
>> Or you can construct (choiceless) Brouwerian counterexamples to your
>> theorem (trivial ones: subsets of a singleton set, for example, which
>> cannot be shown to be either empty or non-empty: such a set, trivially
>> ordered, would be a counter-example to your theorem), but I don't
>> think
>> that is what you are asking for.
>
> It would be a counterexample if it were possible to show that it not
> one
> of the 11 known solutions. But you can't show it differs from the
> empty
> solution.
No you can't but that is not the point of a Brouwerian counterexample:
you can construct a subset of {0}?such that if you know either that the
set is {}, or that it is {0} then you have resolved ... the unsolved
mathematical problem of your choosing. You have therefore established
that it is very unlikely that the theorem stating that there are only
eleven order types will ever be proved, which is all you can hope to
achieve constructively when a result has a classical proof (as
Shipman's "11 types" theorem does in the case of the reals): a
straightforward counterexample is impossible. Are you claiming that the
term "Brouwerian counterexample" is a misnomer?
Regards,
Jeremy Clark
From nate at math.mit.edu Thu Mar 10 01:34:09 2005
From: nate at math.mit.edu (Nate Ackerman)
Date: Thu, 10 Mar 2005 01:34:09 -0500 (EST)
Subject: [FOM] Higher Order Set Theory [Ackermann Set Theory]
In-Reply-To: <1110403541.422f69d53859e@webmail.mit.edu>
References: <1110209960.422c75a87f438@webmail.mit.edu>
<05030808572600.01261@localhost.localdomain>
<1110403541.422f69d53859e@webmail.mit.edu>
Message-ID:
I believe Ackermann set theory was an attempt to create a model of set
theory which could deal with definitions of class of classes, class of
class of classes, ect. As I understand it/think of it (and I am sure there
are people on this list who know more about it than I do), the view was
motivated by the idea that our class of all sets is just an initial
segment of in the hierarchy of the universe of all classes.
And what is more (in some sense) any statement which is true in the
universe should be true in the class of all sets. So, natural models of
Ackermann set theory are (V_\alpha, V_\beta) where V_\alpha is an
elementary substructure of V_\beta (and hence \alpha is an inaccessible).
(It is also worth mentioning though that it has been shown that Ackermann
set theory is equiconsistent with ZF)
Nate
From rbj01 at rbjones.com Thu Mar 10 03:47:40 2005
From: rbj01 at rbjones.com (Roger Bishop Jones)
Date: Thu, 10 Mar 2005 08:47:40 +0000
Subject: [FOM] Higher Order Set Theory
In-Reply-To: <37388b095d12f402eaa5ca055e9d7c8a@xortec.fi>
References: <1110209960.422c75a87f438@webmail.mit.edu>
<05030808572600.01261@localhost.localdomain>
<37388b095d12f402eaa5ca055e9d7c8a@xortec.fi>
Message-ID: <05031008474000.01099@localhost.localdomain>
On Wednesday 09 March 2005 11:26 am, Aatu Koskensilta wrote:
> On Mar 8, 2005, at 10:57 AM, Roger Bishop Jones wrote:
> > Higher order set theory is a great deal less problematic
> > than you make it appear.
>
> In a sense higher order set theory is not problematic at all
> and has a natural axiomatization in Morse-Kelley set theory.
> However, from a conceptual point of view, I believe that
> higher order set theory is *much* more problematic than the
> universe of sets, V alone. The problem is that it's not
> obvious how there could be a determined totality of
> subcollections of V over which the second order quantifiers
> were to range over.
Its not at all obvious to me that the supposition that V
exists is coherent.
Obvious proofs to the contrary spring to mind.
How can something be more problematic than an incoherent
supposition?
The only problem here is with V, the supposition of which
is necessary neither for first order nor for higher order
set theory.
Roger Jones
From sbazerque at gmail.com Thu Mar 10 10:32:43 2005
From: sbazerque at gmail.com (Santiago Bazerque)
Date: Thu, 10 Mar 2005 12:32:43 -0300
Subject: [FOM] Higher Order Set Theory
In-Reply-To: <1110403541.422f69d53859e@webmail.mit.edu>
References: <1110209960.422c75a87f438@webmail.mit.edu>
<05030808572600.01261@localhost.localdomain>
<1110403541.422f69d53859e@webmail.mit.edu>
Message-ID:
On Wed, 9 Mar 2005 16:25:41 -0500
Dmytro Taranovsky wrote:
> The problem is whether higher order statements have any meaning or
> are just symbols on paper.
Does anybody know if ZFC is effectively inseparable? I find this
question interesting because if it were, then if ZFC + { any
axiomatizable extension } is consistent, it is recursively isomorphic to
ZFC (i.e. they would share the same deductive structure) by a result of
Pour-El and Kripke (Fund. Math. LXI p. 142).
This is not merely an interpretation of the extension in ZFC, but a
recursive 1-1 mapping I reinterpreting sentences of our new set theory
onto sentences of ZFC that preserves valid deductions, and a bijective
(non-recursive) mapping J between the classes of models of both theories
such that for every sentence alpha and model M of ZFC+{x},
alpha |= M iff I(alpha) |= J(M).
I am of course not saying that this would imply the "symbols on paper"
thesis, but it is new to me and is making me feel a bit uneasy about the
matter.
Sincerely,
Santiago
ps. The result above would probably still hold for any reasonably
axiomatized extension of the langage of ZFC as well, by the way.
From JoeShipman at aol.com Thu Mar 10 16:54:25 2005
From: JoeShipman at aol.com (JoeShipman@aol.com)
Date: Thu, 10 Mar 2005 16:54:25 -0500
Subject: [FOM] Higher Order Set Theory [Ackermann Set Theory]
Message-ID: <6742A50C.43F6EB3D.0BC565F6@aol.com>
Nate, are you any relation to the original Ackermann, losing
a terminal "n" on Ellis Island?
You write:
****
I believe Ackermann set theory was an attempt to create a model of set
theory which could deal with definitions of class of classes, class of
class of classes, ect. As I understand it/think of it (and I am sure there
are people on this list who know more about it than I do), the view was
motivated by the idea that our class of all sets is just an initial
segment of in the hierarchy of the universe of all classes.
And what is more (in some sense) any statement which is true in the
universe should be true in the class of all sets. So, natural models of
Ackermann set theory are (V_\alpha, V_\beta) where V_\alpha is an
elementary substructure of V_\beta (and hence \alpha is an inaccessible).
(It is also worth mentioning though that it has been shown that Ackermann
set theory is equiconsistent with ZF)
****
OK, so Ackermann set theory is equiconsistent with ZF, but
what is the consistency strength of "there exists (V_\alpha,
V_\beta) where \alpha is inaccessible and V_\alpha is an
elementary substructure of V_\beta" ?
"There are more than continuum many inaccessible cardinals"
certainly works, because then at least two have the same
theory, and the corresponding ranks satisfy that the lower is
an elementary substructure of the higher. If there are
exactly continuum many inaccessible cardinals, it seems one
could apply Easton's theorem to construct a model with this
many inaccessibles where each of the corresponding ranks has
a different first-order theory, so maybe this is an exact
answer, but I need to check that this actually works.
-- JS
From nate at math.mit.edu Thu Mar 10 23:33:31 2005
From: nate at math.mit.edu (Nate Ackerman)
Date: Thu, 10 Mar 2005 23:33:31 -0500 (EST)
Subject: [FOM] Higher Order Set Theory [Ackermann Set Theory]
In-Reply-To: <6742A50C.43F6EB3D.0BC565F6@aol.com>
References: <6742A50C.43F6EB3D.0BC565F6@aol.com>
Message-ID:
Hi Joe,
> OK, so Ackermann set theory is equiconsistent with ZF, but
> what is the consistency strength of "there exists (V_\alpha,
> V_\beta) where \alpha is inaccessible and V_\alpha is an
> elementary substructure of V_\beta" ?
I believe it suffices to have a single inaccessible, although I don't
remember the proof off the top of my head (but I don't think it is that
hard). I also remember seeing a proof of the other direction. That if you
have V_\alpha < V_\beta, then V_\alpha \models ZF, but once again I don't
remember the proof off the top of my head.
> Nate, are you any relation to the original Ackermann, losing
> a terminal "n" on Ellis Island?
Nope, I am not related at all to the original Ackermann,
Nate
From deck at ba-mosbach.de Fri Mar 11 13:35:35 2005
From: deck at ba-mosbach.de (Klaus-Georg Deck)
Date: Fri, 11 Mar 2005 19:35:35 +0100
Subject: [FOM] Antwort: Re: Higher Order Set Theory [Ackermann Set Theory]
In-Reply-To:
Message-ID:
Hi Nate and Joe,
as far as I remember, the proof is due to "Levy & Vaught: Principles of
partial reflection in the set theories of Zermelo and Ackermann, PacJMath
11 (1961), 1045-62.
Klaus-Georg
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Hi Joe,
> OK, so Ackermann set theory is equiconsistent with ZF, but
> what is the consistency strength of "there exists (V_\alpha,
> V_\beta) where \alpha is inaccessible and V_\alpha is an
> elementary substructure of V_\beta" ?
I believe it suffices to have a single inaccessible, although I don't
remember the proof off the top of my head (but I don't think it is that
hard). I also remember seeing a proof of the other direction. That if you
have V_\alpha < V_\beta, then V_\alpha \models ZF, but once again I don't
remember the proof off the top of my head.
> Nate, are you any relation to the original Ackermann, losing
> a terminal "n" on Ellis Island?
Nope, I am not related at all to the original Ackermann,
Nate
_______________________________________________
FOM mailing list
FOM at cs.nyu.edu
http://www.cs.nyu.edu/mailman/listinfo/fom
From enayat at american.edu Fri Mar 11 14:16:02 2005
From: enayat at american.edu (Ali Enayat)
Date: Fri, 11 Mar 2005 14:16:02 -0500
Subject: [FOM] ZFC and Recursive Inseparability
Message-ID: <001301c5266e$c4de7ae0$2e1c4d0c@Home>
This is a reply to a recent query of Santiago Bazerque (Wed, March 9), who
has asked:
>Does anybody know if ZFC is effectively inseparable? I find this
>question interesting because if it were, then if ZFC + { any
>axiomatizable extension } is consistent, it is recursively isomorphic to
>ZFC (i.e. they would share the same deductive structure) by a result of
>Pour-El and Kripke (Fund. Math. LXI p. 142).
The answer is positive, since ZFC "contains enough arithmetic". Here
"enough arithmetic" can be even taken as a finite fragment of Peano
arithmetic, such as the theory known as Robinson's Q.
Regards,
Ali Enayat
From enayat at american.edu Fri Mar 11 15:06:54 2005
From: enayat at american.edu (Ali Enayat)
Date: Fri, 11 Mar 2005 15:06:54 -0500
Subject: [FOM] On full reflection at an inaccessible
Message-ID: <001c01c52675$dfaaf7c0$2e1c4d0c@Home>
This is a reply to Shipman's recent query (March 16), who wrote:
>OK, so Ackermann set theory is equiconsistent with ZF, but
>what is the consistency strength of "there exists (V_\alpha,
>V_\beta) where \alpha is inaccessible and V_\alpha is an
>elementary substructure of V_\beta" ?
1. Let T be the theory described by Shipman in the above paragraph. A strict
upper bound for the consistency strength of T is the consistency strength of
ZF + "there is a Mahlo cardinal". Even better: the latter theory implies the
former theory. This is because for an inaccessible beta, the set of alpha's
below beta such that V(alpha) is an elementary submodel of V(beta) form a
closed unbounded subset of beta [by a routine Skolem hull argument].
2. It is easy to see that T indeed proves the assertion "there is a proper
class of inaccessibles". For otherwise, there would be an upper bound theta
to the inaccessibles in V(alpha), and by elementarity, theta would also
serve as an upper bound to inaccessibles in V(beta), thus contradicting the
inaccessibility of alpha in V(beta).
3. This contradicts the claim of Shipman, where he writes:
>"There are more than continuum many inaccessible cardinals"
>certainly works, because then at least two have the same
>theory, and the corresponding ranks satisfy that the lower is
>an elementary substructure of the higher.
The resolution is as follows: if there are more than continuum many
inaccessibles, then two would surely have the same first order theory, but
this does not guarantee elementarity.
4. Indeed, even the assertion V(beta) is a model of ZFC + "there is a proper
class of inacessibles" is strictly weaker than "V(alpha) is an elementary
submodel of V(beta) and alpha is inaccessible", assuming ,say, that there
exists a Mahlo cardinal, kappa.
To see this, just look at the *first* ordinal beta such that V(beta) is a
model of ZFC + "there is a proper class of inaccessibles" [by reflection
over V(kappa), as in (1), such an ordinal exists]. There is no ordinal
alpha below beta such that V(alpha) is an elementary submodel of V(beta),
for if such an alpha existed, then it would also have to satisfy ZFC +
"there is a proper class of inaccessibles", thereby contradicting the
minimality of beta.
Best regards,
Ali Enayat
From enayat at american.edu Fri Mar 11 16:02:35 2005
From: enayat at american.edu (Ali Enayat)
Date: Fri, 11 Mar 2005 16:02:35 -0500
Subject: [FOM] Full reflection does not imply inaccessibility
Message-ID: <000901c5267d$a8e255a0$b67d4d0c@Home>
In his message of March 10, 2005, Nate Ackerman writes:
>So, natural models ofAckermann set theory are (V_\alpha, V_\beta) where
>V_\alpha is an
>elementary substructure of V_\beta (and hence \alpha is an inaccessible).
The parenthetical statement is false. Indeed, if V(beta) is a model of ZFC
such that beta has uncountable cofinality, then the first alpha such that
V(alpha) is an elementary substructure of V(beta) has countable cofinality,
and is therefore not inaccessible.
To see this, let alpha_n be the first ordinal less than beta such that
V(alpha_n) is a Sigma_n elementary submodel of the universe (such an alpha_n
exists by Levy's version of the Montague-Vaught- reflection theorem). The
desired alpha is the supremum of the alpha_n's.
Regards,
Ali Enayat
From solovay at Math.Berkeley.EDU Fri Mar 11 21:47:27 2005
From: solovay at Math.Berkeley.EDU (Robert M. Solovay)
Date: Fri, 11 Mar 2005 18:47:27 -0800 (PST)
Subject: [FOM] Higher Order Set Theory [Ackermann Set Theory]
In-Reply-To:
Message-ID:
In a recent posting to FOM Joe Shipman writes:
OK, so Ackermann set theory is equiconsistent with ZF, but
what is the consistency strength of "there exists (V_\alpha,
V_\beta) where \alpha is inaccessible and V_\alpha is an
elementary substructure of V_\beta" ?
"There are more than continuum many inaccessible cardinals"
certainly works, because then at least two have the same
theory, and the corresponding ranks satisfy that the lower is
an elementary substructure of the higher.
In reply Nate Ackerman writes:
I believe it suffices to have a single inaccessible, although I don't
remember the proof off the top of my head (but I don't think it is that
hard).
Let us call the principle under discussion SP. [So SP asserts:
"there exists (V_\kappa, V_\beta) where \kappa is inaccessible and
V_\kappa is an elementary substructure of V_\beta"]
In fact the consistency strength of SP is quite a bit larger than
Shipman asserts. I will give fairly sharp upper and lower bounds later in
this letter [they are somewhat technical] but first here are some easily
stated results:
1) In ZFC + SP we can prove the existence of an inaccessible limit
of inaccessibles. {So Shipman's proof must have an error. I will point out
the troublesome step below.}
2) ZFC proves: If alpha is Mahlo, then V_alpha is a model of ZFC +
SP.
I will first prove 1) and 2). Then I will state [and then prove]
the more precise bounds previously alluded to. Then I will take up
Shipman's proof.
Proof of 1): We work in ZFC + SP. First, V_beta thinks there is
an inaccessible. So V_kappa thinks there is an inaccessible. So there is
an inaccessible less than kappa.
Suppose toward a contradiction that kappa is not the sup of the
inaccessibles less than kappa. Let eta be this sup. Then V_kappa thinks
that eta is the sup of the inaccessibles. But then so does V_beta. This is
absurd since kappa is an inaccessible greater than eta known to V_beta.
So kappa is an inaccessible limit of inaccessibles. This proves
1).
We turn to 2). Let alpha be a Mahlo cardinal. Then the set of
gamma < alpha such that V_gamma is an elementary submodel of V_alpha is a
club in alpha. So the set, A, of inaccessible cardinals gamma such that
V_gamma is a proper elementary submodel of V_alpha is stationary in alpha.
Let kappa be the least member of A and beta the second member. Then kappa
and beta instantiate the fact that SP holds in V_alpha. Claim 2) is now
clear.
To sharpen 1) I need to introduce the theory ZM. Roughly ZF:
inaccessible = ZM: Mahlo.
Precisely, ZM will be obtained by adding the following scheme of
axioms to ZFC [one axiom for each n in omega]:
The class of all ordinals alpha such that [(a) alpha is
inaccessible and (b) V_alpha is a Sigma_n elementary submodel of V] is
unbounded in the class OR of all ordinals.
The sharpened form of 1) asserts: Let kappa, beta instantiate SP.
Then V_kappa is a model of ZM.
Let's prove this. Let kappa, beta instantiate SP. Of course,
V_kappa is a model of ZFC. Towards a contradiction let the new axiom with
index n fail in V_kappa. Then certainly, V_beta thinks that kappa is
inaccessible and a Sigma_n elementary submodel of V. So V_kappa thinks
there are such alpha. Let eta be a sup of the set of alpha < kappa which
are inaccessible and give Sigma_n elementary submodels V_alpha. Since the
axiom fails in V_kappa, eta < kappa. But then eta has the same property in
V_beta. This is absurd since kappa is > eta and is inaccessible and with
V_kappa a Sigma_n elementary submodel of V_beta.
For the sharpened form of 2) I need to introduce a slight
strengthening of ZM. Introduce a new predicate Sat. Add the [finitely
many] Tarski axioms which says that Sat codes up the satisfaction relation
for the language of set-theroy. In the various schemes of ZM allow the
predicate Sat to appear. {In particular for the new scheme asserting the
existence of many inaccessilbes interpret "Sigma_n elementary submodel" to
refer to the language with Sat added.}
Call this enlarged theory Sem(ZM). Then there is no difficulty
adapting our proof of 2) to show: Sem(ZM) proves SP.
Finally, the false step in Shipman's discussion is this: Suppose
that kappa < gamma are inaccessibles such that V_kappa and V_gamma have
the same first order theories. Then there is no reason to suppose that
V_kappa is an elementary submodel of V_gamma since the notion of
elementary submodel refers to sentences with parameters chosen from
V_kappa.
--Bob Solovay
From dmytro at MIT.EDU Fri Mar 11 22:44:47 2005
From: dmytro at MIT.EDU (Dmytro Taranovsky)
Date: Fri, 11 Mar 2005 22:44:47 -0500
Subject: [FOM] Higher Order Set Theory [Ackermann Set Theory]
Message-ID: <1110599087.423265af79dd5@webmail.mit.edu>
Joe Shipman wrote:
> what is the consistency strength of "there exists (V_\alpha,
> V_\beta) where \alpha is inaccessible and V_\alpha is an
> elementary substructure of V_\beta" ?
The consistency strength of ZFC with that statement (and beta>alpha; Nate
Ackerman's answer is technically correct) is slightly above ZFC + {there is
Sigma-n correct inaccessible}_n (which is above ZFC + there is a proper class
of inaccessible cardinals), and is below ZFC minus power set plus there is a
Mahlo cardinal.
Being an elementary substructure implies that V(alpha) satisfies correct
statements with parameters in V(alpha). If we only allow formulas without
parameters, then the statement would be stronger than ZFC + there is an
(infinite) ordinal k such that there are at least k inaccessible cardinals and
L_k satisfies ZFC, but weaker than ZFC + there are omega_1 inaccessible
cardinals.
Dmytro Taranovsky
From solovay at Math.Berkeley.EDU Sat Mar 12 18:35:12 2005
From: solovay at Math.Berkeley.EDU (Robert M. Solovay)
Date: Sat, 12 Mar 2005 15:35:12 -0800 (PST)
Subject: [FOM] Higher Order Set Theory [Ackermann Set Theory]
In-Reply-To: <1110599087.423265af79dd5@webmail.mit.edu>
Message-ID:
Mr. Taranovsky's claims about the weakened form of SP are not correct.
Details follow below.
On Fri, 11 Mar 2005, Dmytro Taranovsky wrote:
>
> Being an elementary substructure implies that V(alpha) satisfies correct
> statements with parameters in V(alpha). If we only allow formulas without
> parameters, then the statement would be stronger than ZFC + there is an
> (infinite) ordinal k such that there are at least k inaccessible cardinals and
> L_k satisfies ZFC, but weaker than ZFC + there are omega_1 inaccessible
> cardinals.
>
First let me call SP- the variant of SP where we only require that
"there exists a strongly inaccessible kappa and an alpha > kappa such that
V(alpha) and V(kappa) are elementarily equivalent [satisfy the same
sentences without parameters]".
ZFC + "there exist aleph_2 strongly inaccessible cardinals" proves
the consistency of ZFC + SP-; ZFC + SP- proves the consistency of ZFC +
"the order type of the class of inaccessible cardinals is greater than
aleph_1". Clearly the second claim refutes the last claim of the quoted
passage from Taranovsky.
First work in ZFC + "there exists aleph_2 inaccessible cardinals".
Then in L, there are at least aleph_2 inaccessibles. But in L, GCH holds
and we can use the Shipman argument to prove the consistency of ZFC + SP-.
Since arithmetical statements are absolute from L to V, this completes the
proof of Con(ZFC + SP-).
Now, towards our second claim, work in ZFC + SP-. Clearly SP-
relativizes to L. So without loss of generality, we work from here on out
in the theory ZFC + V=L + SP-. Let kappa be inaccessible and alpha > kappa
such that V(kappa) and V(alpha) are elementarily equivalent.
We seek to show that ZFC + "the order type of the inaccessibles is
greater than aleph_1" holds in L(kappa). If there are at least aleph_2
inaccessibles in V(kappa) we are done. So, wlog, assume that there is an
ordinal eta < aleph_2 which is the order-type of the strong inaccessibles
in V(kappa). Similarly, let eta* be the order-type of the strong
inaccessibles in V(alpha). Then eta < eta* since kappa is strongly
inaccessible in V(alpha).
We first argue that eta is not < aleph_1. If it were, let X be the
eta^{th} subset of omega. Clearly eta* has a similar definition in
V(alpha) and using the elementary equivalence of V(kappa) and V(alpha) we
conclude first that X = X* (X* is the eta*^{th} subset of aleph_1) in
V(alpha)) and then that eta = eta*. Contradiction.
Similarly if eta = aleph_1 in V(kappa) then eta* = aleph_1 in
V(alpha). Since both V(kappa) and V(alpha) both correctly compute aleph_1,
we again have eta = eta* which is absurd.
The upshot is that eta > aleph_1 as was to be shown.
--Bob Solovay
From solovay at Math.Berkeley.EDU Sun Mar 13 03:30:05 2005
From: solovay at Math.Berkeley.EDU (Robert M. Solovay)
Date: Sun, 13 Mar 2005 00:30:05 -0800 (PST)
Subject: [FOM] Higher Order Set Theory [Ackermann Set Theory]
In-Reply-To:
Message-ID:
A tiny correction to my previous posting:
>
> We first argue that eta is not < aleph_1. If it were, let X be the
> eta^{th} subset of omega. Clearly eta* has a similar definition in
> V(alpha) and using the elementary equivalence of V(kappa) and V(alpha) we
> conclude first that X = X* (X* is the eta*^{th} subset of aleph_1) in
> V(alpha)) and then that eta = eta*. Contradiction.
Here I meant to say that X* is the eta*^{th} subset of omega [and
not aleph_1].
--Bob Solovay
From dmytro at MIT.EDU Sun Mar 13 21:56:21 2005
From: dmytro at MIT.EDU (Dmytro Taranovsky)
Date: Sun, 13 Mar 2005 21:56:21 -0500
Subject: [FOM] Question on Second Order Foundations
Message-ID: <1110768981.4234fd55c05c3@webmail.mit.edu>
Ordinary mathematics can be developed in the language of objects and binary
relations. The intended models have the number of objects sufficiently large
relative to smaller numbers and include all binary relations between the
objects.
What is the most elegant way to axiomatize the system?
The conditions on the intended models uniquely fix the theory. The theory has
the same Turing degree as second order set theory; and to some extent, the
languages have the same expressive power.
The axiomatization should be at least as strong as ZFC. One can "transcribe"
NBG or Morse-Kelley set theory into the language, but straightforward attempts
of doing that may not be very elegant. Note that the language allows
quantification over relations, but does not include any particular relations.
Although set-theoretic foundations will remained preferred, looking at the
foundations from a different angle can yield new insights.
Sincerely,
Dmytro Taranovsky
From costaleite at gmail.com Mon Mar 14 07:04:04 2005
From: costaleite at gmail.com (COSTA LEITE Alexandre)
Date: Mon, 14 Mar 2005 13:04:04 +0100
Subject: [FOM] School of Logic in Montreux, Switzerland
Message-ID:
FIRST WORLD SCHOOL ON UNIVERSAL LOGIC
Montreux - Switzerland, March 26-30, 2005
This unique event includes 21 tutorials on general techniques for the
study of logics.
Note that 12 tutorials will be fully available during the Easter
week-end, Saturday
26th - Sunday 27 th - Monday 28th.
The second series of 9 tutorials will be given on Tuesday 29th and
Wednesday 30th.
See the detailed programme on the website www.uni-log.org
LIST OF THE 21 TUTORIALS
1. Combination of Logics - Carlos Caleiro - IST, Portugal
2. Introduction to Universal Logic - Jean-Yves Beziau - SNF, Switzerland
3. Abstract Model Theory - Marta Garcia-Matos - Helsinki, Finland
4. Tableaux Systems - Andreas Herzig - IRIT, France
5. Many-Valued Semantics - Walter Carnielli & Juliana Bueno - UNICAMP, Brazil
6. Abstract Proof Theory - Luiz Carlos Pereira & Ana Teresa Martins -
PUC/UFC, Brazil
7. Adaptive Logics - Diderik Batens & Joke Meheus - Ghent, Belgium
8. Kripke Structures - Darko Sarenac - Stanford, US
9. Category Theory and Logic - Andrei Rodin - ENS, Paris
10. Consequence Operators - Piotr Wojtylak - Katowice, Poland
11. Multiple-Conclusion Logics - Jo?o Marcos - IST, Portugal
12. Nonmonotonic Logics - David Makinson - King's College, London
13. Abstract Algebraic Logic - Josep Maria Font - Barcelona,Spain
14. Substructural Logics - Francesco Paoli - Cagliari, Italy
15. Universal Computation - Roberto Lins - Rio de Janeiro, Brazil
16. Labelled Deductive Systems - Luca Vigano - ETH, Zurich
17. Logics and Games - Jacques Duparc - Lausanne, Switzerland
18. Combinatory Logic and Lambda Calculus - Henri Volken - Lausanne, Switzerland
19. Universal Algebra for Logics - Joanna Grygiel - Czestochowa, Poland
20. Fuzzy Logics - Petr H?jek and Petr Cintula - Academy of Sciences, Prague
21. Logics for semistructured data - Maarten Marx - Amsterdam, The Netherlands
The School will be followed by the
FIRST WORLD CONGRESS ON UNIVERSAL LOGIC
Montreux - Switzerland, March 31- April 3, 2005
Featuring 14 invited spakers (A.Avron, D.Batens, J.Czelakowski, K.Dosen, M.Dunn,
D.Gabbay, R.Jansana, A.Koslow, V.de Paiva, K.Segerberg, J.Vaananen, V.Vasyukov,
Y.Venema, plus a mysterious secret speaker),
and more that 120 contributed talks.
A contest will also happen during this event.
See details at
www.uni-log.org
************************************************************************
Regular registration fees:
Congress only or School only: CHF 280 / USD 250 / Euro 190
Congress + School: CHF 380 / USD 330 / Euro 260
Reduced fees for students or people from countries with low currencies:
Half of the School (26-27-28 or 29-30): CHF 75 / USD 60 / Euro 50
School: CHF 150 / USD 130 / Euro 100
Congress: CHF 150 / USD 130 / Euro 100
Congress + School: CHF 200 / USD 180 / Euro 140
From m.kohlhase at iu-bremen.de Tue Mar 15 05:43:01 2005
From: m.kohlhase at iu-bremen.de (Michael Kohlhase)
Date: Tue, 15 Mar 2005 11:43:01 +0100
Subject: [FOM] Second CfP: MKM 2005 (extended deadline: May 15)
Message-ID: <4236BC35.7030109@iu-bremen.de>
[Please post - apologies for multiple copies.]
============================================
MKM 2005
Fourth International Conference on
MATHEMATICAL KNOWLEDGE MANAGEMENT
http://www.mkm-ig.org/meetings/mkm05/
15. - 16. July 2005 (Workshops: 14. July)
Bremen --- Germany
(organized by International University Bremen)
CALL FOR PAPERS
Mathematical Knowledge Management is a new field in the intersection
of mathematics and computer science.
We need new techniques for managing the enormous volume of
mathematical knowledge available in current mathematical sources and
making it available through the new developments in information
technology. A list of topics (to be understood as specialized to the
realm of mathematical information) comprises but is not restricted to:
Knowledge representation Repositories of formalized mathematics
Metadata Deduction systems
Data mining Computer Algebra Systems
Digital libraries Authoring languages and tools
Searching and retrieving Interactive learning
Languages of mathematics Web presentation of mathematics
Math assistants MathML- and XML-based standards
SUBMISSION
The deadline for submissions is March 20, 2005. Submitted papers should
not exceed 15 pages, must be original and not submitted for publication
elsewhere. All papers submitted to the Conference will be reviewed.
Accepted papers will appear in the proceedings after the Conference.
PROCEEDINGS
We we will publish the post-conference proceedings in the
Springer-Verlag LNAI or LNCS series (which will depend on the
submissions). For instructions see http://www.mkm-ig.org/meetings/mkm05/.
Authors of accepted papers are expected to present their work
at the conference.
IMPORTANT DATES
Submission Deadline: 15. May 2005
Notification of acceptance/rejection: 15. June, 2005
Conference copies: 3. July, 2005
Conference: 15. - 16. July, 200
Affiliated Workshops: 14 July, 2005
Camera-ready Copy: 14. August 2005
Proceedings ship: Mid-October 2005
PROGRAM COMMITTEE
Michael Kohlhase International University Bremen, Germany (Chair)
Andrew Adams Reading University, UK
Andrea Asperti University of Bologna, Italy
Richard Baraniuk Rice University, USA
Christoph Benzmueller Saarland University, Germany
Olga Caprotti University of Helsinki, Finland
Mike Dewar NAG, Ltd., UK
Bill Farmer McMaster University, Canada
Tetsuo Ida University of Tsukuba, Japan
Fairouz Kamareddine Heriott Watt University, Scotland
Andrzej Trybulec University of Bialystok, Poland
Robert Miner Design Science, USA
Till Mossakowski University Bremen, Germany
Stephen Watt University Western Ontario, Canada
RELATED LINKS
MKM 2001, http://www.risc.uni-linz.ac.at/institute/conferences/MKM2001
MKM 2003, http://www.cs.unibo.it/MKM03/
MKM 2004 http://www.mizar.org/MKM2004/
NA-MKM 2002 http://imps.mcmaster.ca/na-mkm-2002/
NA-MKM 2004 http://imps.mcmaster.ca/na-mkm-2004/
To become a member of the MKM Interest group go to the conference or
subscribe to the MKM mailing list at
http://lists.iu-bremen.de/mailman/admin/projects-mkm-ig
--
-------------------------------------------------------------------------
Prof. Dr. Michael Kohlhase, Office: Research 1, Room 62
Professor for Computer Science Campus Ring 12,
School of Engineering & Science D-28758 Bremen, Germany
International University Bremen tel/fax: +49 421 200-3140/-493140
http://www.faculty.iu-bremen.de/mkohlhase
--------------------------------------------------------------------------
From dmytro at MIT.EDU Wed Mar 16 16:51:36 2005
From: dmytro at MIT.EDU (Dmytro Taranovsky)
Date: Wed, 16 Mar 2005 16:51:36 -0500
Subject: [FOM] Question on Second Order Foundations
In-Reply-To: <1110768981.4234fd55c05c3@webmail.mit.edu>
References: <1110768981.4234fd55c05c3@webmail.mit.edu>
Message-ID: <1111009896.4238aa6899962@webmail.mit.edu>
In my previous posting, I noted that almost all of current mathematics can be
formalized in second order logic with the universe having sufficiently many
objects relative to smaller numbers. My best axiomatization is
1. The usual logical rules.
2. The comprehension schema.
3. Uncountability of the universe.
4. Existence of a binary relation enumerating all sets lacking a bijection with
the universe, with every set coded by a single object disjoint from the set.
The fourth axiom includes choice and inaccessibility of the universe. Its
semi-formalization is that there is R such that: for every x, not xRx; not x=y
--> there is u such that not (xRu<-->yRu); every set (represented by a unary
predicate) lacking a bijection with the universe is {y: xRy} for some x.
The axiomatization is at the level of Kelley-Morse set theory. To resolve basic
questions about the number of elements, one may want to add weak compactness
for the universe, which is easily formulated in second order logic through the
partition relation for graphs.
Dmytro Taranovsky
http://web.mit.edu/dmytro/www/main.htm
From fairouz at macs.hw.ac.uk Thu Mar 17 11:26:47 2005
From: fairouz at macs.hw.ac.uk (Fairouz Kamareddine)
Date: Thu, 17 Mar 2005 16:26:47 +0000
Subject: [FOM] ESSLLI 2005 registration Now open
Message-ID:
ESSLLI 2005
17th European Summer School in Logic, Language and Information
The annual summer school of FoLLI,
the Association for Logic, Language and Information.
Heriot-Watt University
Edinburgh, Scotland
8-19 August, 2005
--------------------------
|REGISTRATION IS NOW OPEN|
--------------------------
Go to http://www.macs.hw.ac.uk/esslli05/
and follow the registration page.
(Note, this is during the Edinburgh famous international festival,
so accommodation must be reserved promptly to guarantee accommodation).
The main focus of ESSLLI is on the interface between linguistics,
logic and computation. The school has developed into an important
meeting place and forum for discussion for students, researchers and
IT professionals interested in the interdisciplinary study of Logic,
Language and Information.
ESSLLI courses cover a wide variety of topics within six areas of
interest: Logic, Computation, Language, Logic and Computation,
Computation and Language, Language and Logic.
Foundational courses aim to provide truly introductory courses into a
field. The courses presuppose absolutely no background knowledge. In
particular, they should be accessible to people from other
disciplines.
Introductory courses are intended to equip students and young
researchers with a good understanding of a field's basic methods and
techniques, and to allow experienced researchers from other fields to
acquire the key competences of neighboring disciplines, thus
encouraging the development of a truly interdisciplinary research
community.
Advanced courses are intended to enable participants to acquire more
specialized knowledge about topics they are already familiar with.
Workshops are intended to encourage collaboration and the
cross-fertilization of ideas by stimulating in-depth discussion of
issues which are at the forefront of current research in the field. In
these workshops, students and researchers can give presentations of
their research.
In addition to courses and workshops there are evening lectures, a
student session and a number of satellite events (to be announced
later). The aim of the student session is to provide Masters and PhD
students with an opportunity to present their own work to a
professional audience, thereby getting informed feedback on their own
results. Unlike workshops, the student session is not tied to any
specific theme.
Looking forward to seeing you at ESSLLI 2005 in beautiful Edinburgh
during the impressive Edinburgh international festival (see
http://www.eif.co.uk/festival2005/)
Fairouz Kamareddine
ESSLLI 2005 organising chair
From shuly at cs.haifa.ac.il Fri Mar 18 01:33:05 2005
From: shuly at cs.haifa.ac.il (Shuly Wintner)
Date: Fri, 18 Mar 2005 08:33:05 +0200
Subject: [FOM] FG-MOL 2005: Final Call for Papers
Message-ID: <200503180921.j2I9Lfi31537@imap.science.uva.nl>
*** Our apologies for multiple copies! ***
FG-MOL 2005: The 10th conference on Formal Grammar and
The 9th Meeting on Mathematics of Language
Collocated with the European Summer School in Logic, Language and
Information
Edinburgh, Scotland, 5-7 August 2005
http://www.formalgrammar.tk/
Deadline for paper submission: April 1st, 2005
Call for Papers
Background
FG-MOL 2005 is the 10th conference on Formal Grammar and the 9th
Meeting on the Mathematics of Language, to be held in conjunction with
the European Summer School in Logic, Language and Information, which
takes place in 2005 in Edinburgh.
Previous Formal Grammar meetings were held in Barcelona (1995), Prague
(1996), Aix-en-Provence (1997), Saarbruecken (1998), Utrecht (1999),
Helsinki (2001), Trento (2002), Vienna (2003) and Nancy (2004).
MoL meetings are organized biennially by the Association for
Mathematics of Language, which is a Special Interest Group of the
Association for Computational Linguistics. This is the second time the
two events are held in tandem, following the success of FG-MOL 2001.
Aims and Scope
FG-MOL provides a forum for the presentation of new and original
research on formal grammar, mathematical linguistics and the
application of formal and mathematical methods to the study of natural
language.
Themes of interest include, but are not limited to,
o formal and computational phonology, morphology, syntax, semantics and pragmatics;
o model-theoretic and proof-theoretic methods in linguistics;
o logical aspects of linguistic structure;
o constraint-based and resource-sensitive approaches to grammar;
o learnability of formal grammar;
o integration of stochastic and symbolic models of grammar;
o foundational, methodological and architectural issues in grammar;
o mathematical foundations of statistical approaches to linguistic analysis.
Previous conferences in this series have welcomed papers from a wide variety of frameworks.
Invited Speakers
TBA
Submission Details
We invite electronic submissions of original, unpublished 30-minute
papers (including questions, comments, and discussion). Papers must be
submitted using a dedicated web-based form by April 1, 2005. Papers
should be anonymous and refrain from self-reference. They should be no
longer than 8 pages, single column, point size 11 or 12, written in
English. Submissions should be prepared in plain text (ASCII) or PDF.
Preparation of the manuscript in LaTeX, using the available style
files, is highly recommended. Revised versions will be required to be
in LaTeX, but assistance in translating to LaTeX form will be
available.
Proceedings
Accepted abstracts will be included in the conference proceedings, to
be distributed at the conference. Full, revised versions will be
published after the conference as CSLI Publications Online Proceedings.
Depending on the quality of the papers, we will consider publishing a
selected number of them in a special issue of Research on Language and
Computation.
Social Program
A conference dinner is planned for Saturday, August 6th. More details
will be published in due course.
Important Dates
o April 1st, 2005: Deadline for paper submission
o May 13th, 2005: Notification of acceptance
o May 31st, 2005: Early registration ends
o July 1st, 2005: Full version due
o August 5-7, 2005: Conference dates
Program Committee
Over 40 prominent researchers. See the website for the list.
Organizing committee
o Gerhard Jaeger, University of Bielefeld
o Paola Monachesi, OTS Utrecht
o Gerald Penn, University of Toronto
o James Rogers, Earlham College
o Shuly Wintner, University of Haifa
Sponsors:
The Association for the Mathematics of Language (ACL SigMoL)
Institute for Communicating and Collaborative Systems/Human
Communication Research Centre, University of Edinburgh
The Association for Computational Linguistics (ACL)
Natural Sciences and Engineering Research Council of Canada
http://www.formalgrammar.tk/
From sbuss at math.ucsd.edu Fri Mar 18 13:12:26 2005
From: sbuss at math.ucsd.edu (Sam Buss)
Date: Fri, 18 Mar 2005 18:12:26 -0000
Subject: [FOM] Preprint announcement re Ed Nelson's work
Message-ID: <6.2.0.14.2.20050223224327.0217ae50@math.ucsd.edu>
This is to announce a preprint that relates to some earlier discussions on
the fom group. It is available online at
http://www.math.ucsd.edu/~sbuss/ResearchWeb/nelson/
"Nelson's Work on Logic and Foundations and Other Reflections on
Foundations of Mathematics"
Submitted to Diffusion, Quantum Theory and Radically Elementary Mathematics
(Working title),
edited by W. Faris, Princeton University Press, to appear.
The paper includes discussion of Ed Nelson's philosophies of formalism and
predicative arithmetic, discussion of his unpublished work on automated
theorem proving, and some discussion of my own "definition" of
mathematics. Comments appreciated.
-- Sam Buss
From vladik at cs.utep.edu Fri Mar 18 15:51:45 2005
From: vladik at cs.utep.edu (Vladik Kreinovich)
Date: Fri, 18 Mar 2005 13:51:45 -0700 (MST)
Subject: [FOM] Small types workshop: Constructive analysis,
types and exact real numbers.
Message-ID: <200503182051.j2IKpkZ26018@cs.utep.edu>
FYI.
------------- Begin Forwarded Message -------------
From: Bas Spitters
Small TYPES workshop
"Constructive analysis, types and exact real numbers."
3/4 October 2005
Nijmegen, the Netherlands
The workshop will be held at the campus of the Radboud University Nijmegen
(formerly known as University of Nijmegen or Catholic University of
Nijmegen). This workshop is part of the TYPES project
(http://www.cs.chalmers.se/Cs/Research/Logic/Types/)
Topics include, but are not limited to:
* the development of constructive analysis in type theory
* program extraction from such developments
* exact real number computation
* co-inductive methods for continuous structures
* semantics for real computations (e.g. domain theory, formal topology)
Invited speakers:
Mart?n Escard? and Norbert M?ller
Homepage:
http://www.cs.ru.nl/fnds/typesreal/
Deadline for registration:
September 1st 2005.
We are in the process of investigating the possibilities for a post-workshop
proceedings.
Organizers:
Herman Geuvers
Nicole Messink
Milad Niqui
Bas Spitters
------------- End Forwarded Message -------------
From dmytro at MIT.EDU Sat Mar 19 14:21:36 2005
From: dmytro at MIT.EDU (Dmytro Taranovsky)
Date: Sat, 19 Mar 2005 14:21:36 -0500
Subject: [FOM] Extending Higher Order Set Theory
Message-ID: <1111260096.423c7bc0a0656@webmail.mit.edu>
The language of set theory with reflective ordinals satisfies current
mathematical needs, and it is too early to accept further extensions.
However, the following two related questions may make further study important:
Is there a definable set that is not ordinal definable?
Is it possible to define (the real number coding) first order set theory
locally (without invoking large cardinals)?
An affirmative answer to the second question could radically revise our
understanding of the mathematical universe. It would imply existence of
properties inherently more expressive than those currently accepted.
A reflective ordinal kappa is an ordinal sufficiently large relative to
smaller ordinals for the theory (V, \in, kappa) with parameters in V(kappa) to
be "correct". The idea of reflective ordinals can be iterated and applied to
other extensions of set theory. For example, an ordinal kappa is reflective
in second degree iff the theory (V, \in, kappa, R) with parameters in V(kappa)
is correct, where R is the predicate for reflectiveness. In L, the
indiscernibles are reflective in every finite degree, but non-existence of 0#
in L prevents formalization except for fixed finite degrees.
Higher degree reflective ordinals can be axiomatized inductively using
elementary embeddings analogously to axiomatization of reflective ordinals. In
the models, the critical point kappa (representing Ord) can be reflective in
second degree in j(V(kappa)); j can extend first degree reflectiveness in
V(kappa) until j(kappa); and second degree reflectiveness of kappa can be used
to define such reflectiveness below kappa. (However, it is unclear whether
using a single elementary embedding for all degrees of reflectiveness is
correct.)
A different extension is through reflective sequences. In the most general
form, the predicate (call it P) is defined by (if what is below counts as a
definition):
1. Every set of ordinals that excludes its limit points and has sufficiently
strong reflection properties (for its order type) satisfies P.
2. P(S) iff S is a set of ordinals and for every set of ordinals T of
the same order type satisfying P,
for all x \in V(min(S, T)), Th(V, \in, S, x) = Th(V, \in, T, x).
The predicate for reflective ordinals of finite degree n has slightly greater
expressive power than the predicate for reflective n-tuples of ordinals. For
example, {reflective ordinal of second degree, a larger reflective ordinal}
is a reflective pair. However, infinite reflective sequences have greater
expressive power than iterations of reflectiveness: If S is such a sequence,
then for every ordinal alpha, reflectiveness of degree alpha restricted to
min(S) is definable from S and alpha. It is unclear if infinite reflective
sequences exist and how to axiomatize them.
For a fixed natural number n and ordinal alpha, reflectiveness for n-tuples
below alpha is ordinal definable (this also applies to L). However, could it
be that a real number is ordinal definable iff it is definable from a
reflective n-tuple of ordinals?
The way to proceed may be study analogues of these notions for hereditarily
countable sets since the theory of such sets is well-understood. Will such
studies lead to a local definition of first order set theory?
Additional background information can be found in my postings "Higher Order
Set Theory" and "Axiomatizing Higher Order Set Theory", as well as my paper:
http://web.mit.edu/dmytro/www/NewSetTheory.htm
Dmytro Taranovsky
From martin at eipye.com Sun Mar 20 15:07:05 2005
From: martin at eipye.com (Martin Davis)
Date: Sun, 20 Mar 2005 12:07:05 -0800
Subject: [FOM] moderator away
Message-ID: <5.1.0.14.2.20050320120454.0159ca80@mail.eipye.com>
I'll be at a conference this week, and at a family function next weekend.
Expect FOM service to be slow.
Martin
From Andrzej.Murawski at comlab.ox.ac.uk Mon Mar 21 05:25:37 2005
From: Andrzej.Murawski at comlab.ox.ac.uk (Andrzej Murawski)
Date: Mon, 21 Mar 2005 10:25:37 +0000 (GMT)
Subject: [FOM] CSL'05 Final Call for Papers
Message-ID:
CALL FOR PAPERS
CSL'05, University of Oxford, 22-25 August 2005
http://web.comlab.ox.ac.uk/oucl/conferences/CSL05/
THE EVENT
Computer Science Logic (CSL) is the annual conference of the European
Association for Computer Science Logic (EACSL). The 14th Annual
Conference (and 19th International Workshop), CSL2005, will take
place in the week 22 - 25 August 2005; it will be organised by the
Computing Laboratory at the University of Oxford.
SCOPE
The conference is intended for computer scientists whose research
activities involve logic, as well as for logicians working on issues
significant for computer science. Suggested topics of interest
include: automated deduction and interactive theorem proving,
constructive mathematics and type theory, equational logic and term
rewriting, modal and temporal logic, model checking, logical aspects
of computational complexity, finite model theory, computational proof
theory, logic programming and constraints, lambda calculus and
combinatory logic, categorical logic and topological semantics,
domain theory, database theory, specification, extraction and
transformation of programs, logical foundations of programming
paradigms, linear logic, higher-order logic.
INVITED SPEAKERS
Matthias Baaz (U. of Technology, Vienna)
Ulrich Berger (U. of Wales, Swansea)
Maarten Marx (U. of Amsterdam)
Anatol Slissenko (Université Paris 12)
SUBMISSION
The proceedings will be published in the Springer Lecture Notes in
Computer Science. Papers accepted by the Programme Committee must be
presented at the conference by one of the authors, and final copy
prepared according to Springer's guidelines.
Submitted papers must be in Springer's LNCS style and of no more than
15 pages, presenting work not previously published. They must not be
submitted concurrently to another conference with refereed
proceedings. The PC chair should be informed of closely related work
submitted to a conference or journal by 1 April 2005. Papers authored
or coauthored by members of the Programme Committee are not allowed.
Submitted papers must be in English and provide sufficient detail to
allow the programme committee to assess the merits of the paper. Full
proofs may appear in a technical appendix which will be read at the
reviewer's discretion. The title page must contain: title and
author(s), physical and e-mail addresses, identification of the
corresponding author, an abstract of no more than 200 words, and a
list of keywords.
IMPORTANT DATES
Deadline for abstracts 25 March, 2005
Deadline for papers 1 April, 2005
Notification 15 May, 2005
Final versions due 1 June, 2005
ACKERMANN AWARD
The EACSL Board has decided to launch the Ackermann Award: The EACSL
Outstanding Dissertation Award for Logic in Computer Science. The first
awards will be presented to the recipients at CSL'05. Further
details of the Award can be found at
http://www.dimi.uniud.it/~eacsl/award.html
PROGRAMME COMMITTEE
Albert Atserias (Universitat Politècnica de Catalunya)
David Basin (Eidgenössische Technische Hochschule Zürich)
Martin Escardo (U. of Birmingham)
Zoltan Esik (U. of Szeged)
Martin Grohe (Humboldt-Universitat zu Berlin)
Ryu Hasegawa (U. of Tokyo)
Martin Hofmann (Ludwig-Maximilians-Universität München)
Ulrich Kohlenbach (Darmstadt U. of Technology)
Orna Kupferman (Hebrew U. of Jerusalem)
Paul-Andre Mellies (CNRS / Université Paris 7)
Aart Middeldorp (U. of Innsbruck, Austria)
Dale Miller (INRIA / Ecole Polytechnique)
Damian Niwinski (U. of Warsaw)
Peter O'Hearn (Queen Mary, U. of London)
Luke Ong (U. of Oxford, Chair)
Alexander Rabinovich (U. of Tel Aviv)
Thomas Schwentick (Philipps-Universität Marburg)
Alex Simpson (U. of Edinburgh)
Nicolai Vorobjov (U. of Bath)
Andrei Voronkov (U. of Manchester)
From holmes at diamond.boisestate.edu Mon Mar 21 15:06:50 2005
From: holmes at diamond.boisestate.edu (Randall Holmes)
Date: Mon, 21 Mar 2005 13:06:50 -0700
Subject: [FOM] NFU book by Holmes now available online
Message-ID: <200503212006.j2LK6ouW008969@diamond.boisestate.edu>
Dear members of the FOM and New Foundations lists,
My book "Elementary Set Theory with a Universal Set" has gone out of print,
but I now have permission from my publisher to post a revised version online.
A provisional version is already posted on my web page, but I'm planning
to do a more careful revision (the one which has been posted was prepared
several years ago).
If any readers of this book can bring errors and infelicities to my attention
at this point, this would be effective (they would have a chance of being
corrected!)
Sincerely, Randall Holmes
PS my page is http://math.boisestate.edu/~holmes
From drm39 at cam.ac.uk Tue Mar 22 05:22:15 2005
From: drm39 at cam.ac.uk (D.R. MacIver)
Date: 22 Mar 2005 10:22:15 +0000
Subject: [FOM] Dependent Choice and Ultrafilters
Message-ID:
Does anyone happen to know if ZF + DC(omega) + `Every filter on a set is
contained in an ultrafilter' implies DC(omega_1)? I can't imagine that it
does, but I also can't find any references for it (and my knowledge of
forcing is still rather theoretical and extremely weak, so I can't manage
much in the way of an independence proof myself).
David
From carlos at science.uva.nl Mon Mar 28 03:58:02 2005
From: carlos at science.uva.nl (Carlos Areces)
Date: Mon, 28 Mar 2005 10:58:02 +0200
Subject: [FOM] ESSLLI 2005 - Registration now Open!
Message-ID: <20050328085156.GB4170@remote.science.uva.nl>
ESSLLI 2005
17th European Summer School in Logic, Language and Information
The annual summer school of FoLLI,
the Association for Logic, Language and Information.
Heriot-Watt University
Edinburgh, Scotland
8-19 August, 2005
--------------------------
|REGISTRATION IS NOW OPEN|
--------------------------
Go to http://www.macs.hw.ac.uk/esslli05/
and follow the registration page at:
http://www.macs.hw.ac.uk/esslli05/give-page.php?17
(Note, this is during the Edinburgh famous international festival,
so accommodation must be reserved promptly to guarantee accommodation).
The main focus of ESSLLI is on the interface between linguistics,
logic and computation. The school has developed into an important
meeting place and forum for discussion for students, researchers and
IT professionals interested in the interdisciplinary study of Logic,
Language and Information.
ESSLLI courses cover a wide variety of topics within six areas of
interest: Logic, Computation, Language, Logic and Computation,
Computation and Language, Language and Logic.
Foundational courses aim to provide truly introductory courses into a
field. The courses presuppose absolutely no background knowledge. In
particular, they should be accessible to people from other
disciplines.
Introductory courses are intended to equip students and young
researchers with a good understanding of a field's basic methods and
techniques, and to allow experienced researchers from other fields to
acquire the key competences of neighboring disciplines, thus
encouraging the development of a truly interdisciplinary research
community.
Advanced courses are intended to enable participants to acquire more
specialized knowledge about topics they are already familiar with.
Workshops are intended to encourage collaboration and the
cross-fertilization of ideas by stimulating in-depth discussion of
issues which are at the forefront of current research in the field. In
these workshops, students and researchers can give presentations of
their research.
In addition to courses and workshops there are evening lectures, a
student session and a number of satellite events (to be announced
later). The aim of the student session is to provide Masters and PhD
students with an opportunity to present their own work to a
professional audience, thereby getting informed feedback on their own
results. Unlike workshops, the student session is not tied to any
specific theme.
Looking forward to seeing you at ESSLLI 2005 in beautiful Edinburgh
during the impressive Edinburgh international festival (see
http://www.eif.co.uk/festival2005/)
Fairouz Kamareddine (ESSLLI 2005 organising chair) and
FOLLI (the Association for Logic, Language and Information)
From addison at Math.Berkeley.EDU Tue Mar 29 23:02:17 2005
From: addison at Math.Berkeley.EDU (J. W. Addison)
Date: Tue, 29 Mar 2005 20:02:17 -0800 (PST)
Subject: [FOM] Seventeenth Annual Alfred Tarski Lectures
Message-ID: <200503300402.j2U42HSh013791@yuban.math.berkeley.edu>
The University of California, Berkeley
announces
THE SEVENTEENTH ANNUAL ALFRED TARSKI LECTURES
by
ZLIL SELA
Professor of Mathematics
Hebrew University of Jerusalem
* * *
THE ELEMENTARY THEORY OF A FREE GROUP
Monday, April 4, 2005
4:10 p.m.
60 Evans Hall
VARIETIES OVER FREE GROUPS
Wednesday, April 6, 2005
4:10 p.m.
3 Evans Hall
AE SENTENCES AND QUANTIFIER ELIMINATION
Friday, April 8, 2005
4:10 p.m.
60 Evans Hall
From freek at cs.ru.nl Wed Mar 30 07:52:43 2005
From: freek at cs.ru.nl (Freek Wiedijk)
Date: Wed, 30 Mar 2005 14:52:43 +0200 (MEST)
Subject: [FOM] PhD position in formal mathematics at RU Nijmegen
Message-ID: <200503301252.j2UCqhwa005376@leto.cs.kun.nl>
Subject: PhD position in formal mathematics at RU Nijmegen
In informal calculus, and therefore also in computer algebra
of the Maple/Mathematica kind, one encounters formulas that
only have a "fuzzy" mathematical meaning, like:
ln(0) = -infinity
integral(1/x,x) = ln(x) + C
ln(1+x) = x + O(x^2)
ln(z) = 2.3025850929... log(z)
Ln(z) = ln(z) + 2 k pi i, with k in Z
In particular it is not clear at all what the "="
signs in these formulas mean. The "lack of semantics"
for this kind of formulas is a real source of problems in
computer algebra systems. It regularly causes the formula
manipulation algorithms in those programs to get confused
and give wrong or meaningless answers.
The F.E.A.R. project ("Formalizing Elementary Analysis
Rigorously") has been started to address this problem. The
core activity of this project will be the formalization of a
section of Abramowitz & Stegun using the proof assistant HOL.
The _proofs_ of the statements from that section will _not_
be formalized, but all the definitions that are needed to
phrase those statements will be provided in full detail.
The main aim of the project is to find formal versions
of those statements that are similar in "look and feel"
to their informal counterparts as much as possible.
The intended outcome of the F.E.A.R. project will be a term
language that (i) will enable users of proof assistants
to formalize calculus in a way that is much closer to
informal calculus than it is today (and the aim is to have
the approach be abstract enough to be usable in all proof
assistants and not only in HOL), and that (ii) will allow a
term-rewriting based implementation of calculus in computer
algebra that combines usefulness with a rigorous semantics.
For the details of the F.E.A.R. project, see
or
.
For an approach for dealing with infinity in the formulas
of informal calculus, see
or
. The research
of the F.E.A.R. project can be seen as a extension of the
topic of that paper.
WE ARE LOOKING FOR
a student with a completed Master's degree in mathematics
or computer science (preferably with a thorough knowledge of
complex function theory), who combines a strong mathematical
ability with an active interest in the computerization of
mathematics. Experience with a proof assistant or with the
application of computer algebra to calculus will be a bonus.
WE ARE OFFERING
employment as a PhD student for four years in the group
of Henk Barendregt and Herman Geuvers (see the web page at
), which is one of the foremost
research groups in formal mathematics in the world. We have
a very enthusiastic and international team and participate
in various national and international research projects
(like Types, Calculemus, MoWGLI, MKM and Diamant.) The PhD
position will be combined with an educational program,
and has a salary that is standard for PhD students in the
Netherlands (starting at 2,179 euro/month before taxes,
and growing to 2,517 euro/month max.)
For more information contact Freek Wiedijk .
Reactions should be sent before July 1st, 2005.
From geoff at cs.miami.edu Thu Mar 31 08:26:26 2005
From: geoff at cs.miami.edu (geoff@cs.miami.edu)
Date: Thu, 31 Mar 2005 08:26:26 -0500 (EST)
Subject: [FOM] LPAR-12 in Jamaica
Message-ID: <20050331132626.9764BAF9C@sherman.cs.miami.edu>
--------------------------------------------------------------------------
LPAR-12 Montego Bay, Jamaica
http://www.lpar.net/2005 2nd-6th December 2005
Call For Papers
The 12th International Conference on Logic for Programming Artificial
Intelligence and Reasoning (LPAR-12) will be held 2nd-6th December 2005,
at the Wexford Hotel, Montego Bay, Jamaica. Submission of papers for
presentation at the conference is now invited. Topics of interest include:
+ automated reasoning + propositional reasoning
+ interactive theorem proving + description logics
+ proof assistants + modal and temporal logics
+ proof planning + nonmonotonic reasoning
+ proof checking + constructive logic and type theory
+ rewriting and unification + lambda and combinatory calculi
+ software and hardware verification + logic programming
+ network and protocol verification + constraint programming
+ systems specification and synthesis + logical foundations of programming
+ model checking + computational interpretations of logic
+ proof-carrying code + logic and computational complexity
+ logic and databases + logic in artificial intelligence
+ reasoning over ontologies + knowledge representation and reasoning
+ reasoning for the semantic web + reasoning about actions
Full and short papers are welcome. Full papers may be either regular papers
containing new results, or experimental papers describing implementations
or evaluations of systems. Short papers may describe work in progress or
provide system descriptions. Submitted papers must be original, and not
submitted concurrently to a journal or another conference.
The full paper proceedings of LPAR-12 will be published by Springer-Verlag
in the LNAI series. Authors of accepted full papers will be required to
sign a form transferring copyright of their contribution to Springer-Verlag.
The short paper proceedings of LPAR-12 will be published by the conference.
Submission Instructions
-----------------------
Papers must be prepared using the Springer-Verlag instructions for authors
(http://www.springer.de/comp/lncs/authors.html). Papers may be up to 15
pages. If proofs do not fit in 15 pages, an appendix with proofs may be
added. Short papers may be up to 5 pages. Papers must be submitted in plain
postscript or PDF format, through the online submission system
(http://www.easychair.org/LPAR-05/submit/).
Dates and Deadlines:
+ Submission of full paper abstracts 11th July
+ Submission of full papers 18th July
+ Notification of acceptance of full papers 12th September
+ Camera ready versions of full papers due 3rd October
+ Submission of short papers 26th September
+ Notification of acceptance of short papers 24th October
+ Camera ready versions of short papers due 7th November
Questions related to submission may be sent to the program chairs, Geoff
Sutcliffe and Andrei Voronkov.
--------------------------------------------------------------------------
Jamaica ... Land of LPAR and Reggae
--------------------------------------------------------------------------
**