[FOM] 743: Philosophy of Incompleteness/5
Harvey Friedman
hmflogic at gmail.com
Mon Jan 9 14:32:36 EST 2017
In the previous postings in this series, we discussed, among other things,
A. An existing mathematical question that is widely known and of wide
interest, is shown to be neither provable nor refutable in ZFC.
Has not happened, and not likely to happen.
L CONJECTURE/1. Every mathematical statement presented before 2017 by
a normal mathematician, that involves only sets in the first
omega+omega levels of the cumulative hierarchy, is provable or
refutable in ZFC + V = L.
L CONJECTURE/2. Every mathematical statement presented before 2017 by
a normal mathematician is provable or refutable in ZFC + V = L +
"there is no strongly inaccessible cardinal".
L CONJECTURE/3. Every mathematical statement presented before 2017 by
a normal mathematician is provable or refutable in ZFC + V = L +
"there are infinitely many inaccessible cardinals".
V = L EQUIVALENT/1. V=L(1eq,n). Every definable transitive class under
epsilon, that is not a set, and forms a model of those axioms of ZF
with at most n occurrences of symbols, not counting parentheses,
includes all sets.
V = L EQUIVALENT/2. V=L(2eq,n). Every definable class under epsilon,
that is not a set, and forms a model of those axioms of ZF with at
most n occurrences of symbols, not counting parentheses, satisfies the
same sentences as the class of all sets under epsilon.
MERITS. It obliterates all independence results from ZF in the sense
of the three L Conjectures. Mathematicians generally find independence
unwanted nuisances, even threatening to their personal relationship
with mathematics, and generally approve of the idea of getting rid of
them. That is, getting rid of them if it doesn't affect their work and
the mathematics that they know and care about. Furthermore, the
mathematicians have never encountered any mathematical object that
they had any reason to believe was outside L. In the case of
encountered countable objects, they know or can be made aware of that
these are demonstrably in L. They have heard from set theorists that V
= L precedes the existence of medium to large large cardinals, and
that is a big negative for them. For the mathematicians, that is a big
positive for them, as such enormous objects are very much out there,
and not having to worry about them is a really good thing. Not having
to think about independence and not having to think about medium/large
large cardinals is an attractive combination.
DEMERITS. Why would one believe that all sets are in fact
constructible, or even that all real numbers are constructible? By
comparison, the axioms of ZFC look obvious. And if we assume that even
all real numbers are constructible, then maybe this throws away real
numbers that I am talking about when I prove something about all real
numbers. Also, I hear that set theorists categorically reject V = L so
shouldn't we defer to them on this matter, or at least not go against
what they advise and think? OK, it is more convenient to have V = L so
as to get rid of weird stuff we don't care about or care much about
anyway, but there is TRUTH, and you can't disrespect TRUTH for
convenience.
JOHN STEEL writes:
"Harvey writes:
For the mathematicians, that is a big
positive for them [ that V= L precludes large cardinals] ,
as such enormous objects are very much out there,
and not having to worry about them is a really good thing. Not having
to think about independence and not having to think about medium/large
large cardinals is an attractive combination.
This is a pretty cynical view of "mathematicians". Of course,
it's rational to withhold judgment on what you don't understand.
But rational people also realize that their personal understanding of
a theory removed from their own work is not much of a test of its truth.
And they don't hope that it's wrong because they don't work on it.
I don't understand String Theory, but I don't hope that
it's wrong."
Here is my reading of mathematicians. They generally have a rather
fluid relationship with any concept of absolute truth when it comes to
general mathematical conceptions and general conceptions outside
mathematics. They don't have any clear conception of arbitrary sets
and arbitrary ordinals, and realize that for any mathematics that they
are interested in or care about, such general concepts do not arise.
They do realize that one can of course prove things about such general
notions based on very few principles. But they put it in the general
category of abstract nonsense, like that real useful area "abstract
nonsense" which is essentially general category theory. I.e.,
arbitrary sets and arbitrary ordinals are judged according to how
useful they are for the kind of mathematics they value intrinsically.
E.g., every field whatsoever has a unique algebraic closure is still
taught in graduate school math, although on reflection, I am not quite
sure that is the case anymore (does anybody know?).
Now John and I AGREE that the mathematicians are DEAD WRONG on many
levels. The real difference between John and I is that whereas John
thinks that they are dead wrong because of the developments, post
Goedel, due to John and his close colleagues, through Projective
Hierarchy Incompleteness (being completed by large cardinal
hypotheses), think they are dead wrong, post Goedel, entirely because
of me (with some early help in the 1960's,70's from Tony and Leo)
through Concrete Mathematical Incompleteness (being completed by large
cardinal hypotheses).
Also whereas John seems to think that String Theory is a legitimate
subject in the sense of making claims that have a definite truth
value. Whereas some physicists (maybe most for all I know) do not
accept that in the absence of the possibility of experimental
confirmation and testing, which many (maybe most for all I know)
doubt. Most mathematicians do not (at least readily) accept the same
for arbitrary coherently readable set theoretic statements.
An important suggestion that John is raising is that mathematicians
might take a different view on these matters if they seriously studied
work in abstract set theory. E.g., that statements like V = L will be
perceived to have a definite truth value, and that in the case of V =
L, that true value is F. This I have my doubts about, although it
becomes a chicken and the egg problem. They don't want to seriously
study this work because they have their doubts about not only its
absolute truth, but also their doubts about whether it is useful for
anything that they intrinsically value.
Of course, I claim to be working up a definite resolution of this
impasse, at least in part. Whereas nothing that I do in Concrete
Incompleteness would even suggest absolute truth for general
coherently stated set theoretic statements, it is aimed at
establishing the crucial irremovable usefulness of some abstract set
theoretic statements for intrinsically important mathematics. It is of
course much too early to see how much more convincing these advances
will become, and how they are going to be perceived by the general
mathematical community.
To give a social indicator of how far the general mathematics
community has drifted away from what they generally refer to as
"logic", I have been invited to come and give some talks at U Texas,
Austin later in the year. Looking at the large number of faculty at
the Math Dept there on their website where they list their main
research interests, not even ONE of them lists logic or math logic or
f.o.m. On the other hand, looking at the faculty at the Philosophy
Dept there on the corresponding website, fully 12 out of 39 listed
logic or philosophical logic (or Russell/Frege). This is not a good
place to go much further with this point.
In the next posting I want to talk about strategies in light of the
negative prospects for A at the top of this posting.
************************************************************************
My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ
This is the 743rd in a series of self contained numbered
postings to FOM covering a wide range of topics in f.o.m. The list of
previous numbered postings #1-699 can be found at
http://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/
700: Large Cardinals and Continuations/14 8/1/16 11:01AM
701: Extending Functions/1 8/10/16 10:02AM
702: Large Cardinals and Continuations/15 8/22/16 9:22PM
703: Large Cardinals and Continuations/16 8/26/16 12:03AM
704: Large Cardinals and Continuations/17 8/31/16 12:55AM
705: Large Cardinals and Continuations/18 8/31/16 11:47PM
706: Second Incompleteness/1 7/5/16 2:03AM
707: Second Incompleteness/2 9/8/16 3:37PM
708: Second Incompleteness/3 9/11/16 10:33PM
709: Large Cardinals and Continuations/19 9/13/16 4:17AM
710: Large Cardinals and Continuations/20 9/14/16 1:27AM
700: Large Cardinals and Continuations/14 8/1/16 11:01AM
701: Extending Functions/1 8/10/16 10:02AM
702: Large Cardinals and Continuations/15 8/22/16 9:22PM
703: Large Cardinals and Continuations/16 8/26/16 12:03AM
704: Large Cardinals and Continuations/17 8/31/16 12:55AM
705: Large Cardinals and Continuations/18 8/31/16 11:47PM
706: Second Incompleteness/1 7/5/16 2:03AM
707: Second Incompleteness/2 9/8/16 3:37PM
708: Second Incompleteness/3 9/11/16 10:33PM
709: Large Cardinals and Continuations/19 9/13/16 4:17AM
710: Large Cardinals and Continuations/20 9/14/16 1:27AM
711: Large Cardinals and Continuations/21 9/18/16 10:42AM
712: PA Incompleteness/1 9/2316 1:20AM
713: Foundations of Geometry/1 9/24/16 2:09PM
714: Foundations of Geometry/2 9/25/16 10:26PM
715: Foundations of Geometry/3 9/27/16 1:08AM
716: Foundations of Geometry/4 9/27/16 10:25PM
717: Foundations of Geometry/5 9/30/16 12:16AM
718: Foundations of Geometry/6 101/16 12:19PM
719: Large Cardinals and Emulations/22
720: Foundations of Geometry/7 10/2/16 1:59PM
721: Large Cardinals and Emulations//23 10/4/16 2:35AM
722: Large Cardinals and Emulations/24 10/616 1:59AM
723: Philosophical Geometry/8 10/816 1:47AM
724: Philosophical Geometry/9 10/10/16 9:36AM
725: Philosophical Geometry/10 10/14/16 10:16PM
726: Philosophical Geometry/11 Oct 17 16:04:26 EDT 2016
727: Large Cardinals and Emulations/25 10/20/16 1:37PM
728: Philosophical Geometry/12 10/24/16 3:35PM
729: Consistency of Mathematics/1 10/25/16 1:25PM
730: Consistency of Mathematics/2 11/17/16 9:50PM
731: Large Cardinals and Emulations/26 11/21/16 5:40PM
732: Large Cardinals and Emulations/27 11/28/16 1:31AM
733: Large Cardinals and Emulations/28 12/6/16 1AM
734: Large Cardinals and Emulations/29 12/8/16 2:53PM
735: Philosophical Geometry/13 12/19/16 4:24PM
736: Philosophical Geometry/14 12/20/16 12:43PM
737: Philosophical Geometry/15 12/22/16 3:24PM
738: Philosophical Geometry/16 12/27/16 6:54PM
739: Philosophical Geometry/17 1/2/17 11:50PM
740: Philosophy of Incompleteness/2 1/7/16 8:33AM
741: Philosophy of Incompleteness/3 1/7/16 1:18PM
742: Philosophy of Incompleteness/4 1/8/16 3:45AM
Harvey Friedman
More information about the FOM
mailing list