FW: [FOM] As to strict definitions of potential and actual infinities.

[Alexander Zenkin]

The response is inserted into the text of the Dean Buckner's 'Original
Message' below.

Further (for short):  Dean Buckner = DB:, Alexander Zenkin = AZ:


-----Original Message-----
From: fom-admin at cs.nyu.edu [mailto:fom-admin at cs.nyu.edu] On Behalf Of
Dean Buckner
Sent: Wednesday, January 01, 2003 8:39 PM
To: fom
Subject: Re: [FOM] As to strict definitions of potential and actual
infinities.

DB:
Alexander Zenkin's posting only confirms my view that expressions like
"actual infinity" "potential infinity" "completed infinity" and the like
are vague, imprecise, incoherent &c.

AZ: 
In my FOM-message "As to strict definitions of potential and actual
infinities" (on 31 December, 2002, 12:45) the definitions are offered
just in order to avoid the traditionally vague, imprecise, incoherent,
intuitive, philosophical interpretations of the expressions like "actual
infinity" and "potential infinity". Just therefore the definitions
comprise no one vague, intuitive, philosophical term and "non-logical
elements". Of course, except for some illustrative quotations of
Cantor's quite vague, intuitive 'definitions' of the 'actual infinite'
notion.
	I believe that the Aristotle-Peano's series

	1,2,3, . . . (*)

has a 'physics' not more than the Cantor's transfinite series

	'omega'+1, 'omega'+2, 'omega'+3, . . . (C)

	I suspect that you have no 'physical' pretensions to Cantor's
transfinite series (C). If so, then why you ascribe a 'physical' sense
to the infinite series (*) ?
	Of course you can interpret the series (*) (as well as the
series (C)) as a sequence of temporal steps to accomplish every
operation '+1', e.g., one second, two seconds, three seconds, etc., and
then conclude that the process of the construction of ALL natural
numbers in (*) will demand an infinite time, but the last is impossible
and, consequently, the series (*) can't be completed, i.e., it can't be
actually infinite. However, I don't regard this traditional objection
against actual infinity as a convincing one and use it never and
nowhere. By the way, my ALGIRITHMIC definitions of the concepts of the
potential and actual infinities have no relation to a vague
philosophical 'problem' how many time the presented computer program
generating the series (*) will work; the question is only about the
algorithmical effectiveness of the process: whether the program (the
corresponding Turing machine, if you please) will arrive (will have the
property 'to have . . .', if you please) its halting state, or will not.



DB:
The problem is the dependence on temporal notions or physical processes
(Aristotle's discussions focus either on cutting things up into parts,
or for waiting for things to be completed).  Thus non-logical elements
obtrude into what should be a perfectly logical definition.

AZ: 
The expressions like "arrives at" [a HALTING state] and [a computer
program] "continues [its work] [for]ever", i.e., not arriving at its
HALTING state, also have no relation to a "vague philosophy", since they
are well defined in the theory of algorithms as well as in the Turing
machine theory. I believe that constructive, algorithmic definitions of
the potential and actual infinities concepts in foundations of
mathematics (as well as in set theory) are not less legitimate than
"perfectly logical definitions" which you mean.


DB:
Is it possible for Alexander to restate his definition without using
expressions like "arrives at" "continue [for]ever" and such like?


AZ: 
The expressions like "arrives at" [a HALTING state] and [a computer
program] "continues [its work] [for]ever", i.e., not arriving at its
HALTING state, also have no relation to a "vague philosophy", since they
are well defined in the theory of algorithms as well as in the Turing
machine theory. I believe that constructive, algorithmic definitions of
the potential and actual infinities concepts in foundations of
mathematics (as well as in set theory) are not less legitimate than
"perfectly logical definitions" which you mean.

DB:
And there is nothing _obviously_ wrong with the non-naive definition of
actual infinity, which (as I understand it) simply postulates the
existence of a set of which every object in Zenkin's series is a member.
There is no contradiction in every member n of the series being such
that there exists another member n+1, and that every such object belongs
to the set, which is not itself a member of the series.  And according
to set theory such a set is itself a number (but not a finite one).
There is no appeal to time or process or becoming in this formulation.
On the face of it, there is no contradiction

AZ: 
The postulation of the existence of a well-ordered set (series) (*)
tells us nothing as to whether the infinite set (*) is actual or
potential: Aristotle states (ACCEPTS, see below) that the infinite set
(*) is potential, Cantor states (ACCEPTS, see below) that the same set
(*) is actual. And you are right: stating that the sequence (*) is a
(minimal transfinite) "invariable in all its parts" NUMBER, Cantor does
not "appeal to time or process or becoming", he (and modern set theory)
simply postulates that the infinite series (*) is completed, i.e.,
actual. Without any proof, i.e., as a hidden axiom (see below).

DB:
As John Mayberry has said, if there is something wrong with set theory,
which has been the subject of intense research over the last 100 years,
then it is nothing gross or obvious.

AZ: 
Some before 1543 A.D., a lot of quite advanced scientists stated as
well: "if there is something wrong with Ptolemaic geocentric system,
which has been the subject of intense research over the last 1400 years,
then it is nothing gross or obvious." You see well that a number of
scientists defending the actual infinity concept and the 100-year's
duration of the defence can't be an argument to prove the concept. By
the way, among the scientists who stated that "Infinitum actu non
datur", i.e., who rejected and reject today the actual infinity on which
Cantor's (as well as all modern non-naive) set theory is based, were
Aristotle, Leibniz, Berkeley, Locke, Descartes, Kant, d'Espinosa,
Lagrange, Gauss, Lobachevsky, Cauchy, Kronecker, Hermite, Poincare,
Bair, Borel, Brouwer, Quine, Wittgenstein, Weyl, Luzin, and today -
Feferman, Peregrin, Turchin, and a lot of other outstanding creators of
the classical (i.e., working really) logic and the classical (i.e.,
working really) mathematics as a whole ! Of course, the list itself is
not a proof of the self-contradiction of actual infinity. But it is a
good reason to think deeply of the problem. Just therefore we are needed
now in a strict logical and algorithmical definition of the concepts of
potential and actual infinities without any reference to vague,
intuitive, philosophical arguments such as 'temporal' notions and
'physical' backgrounds of the concepts.

DB:
More subtle is Wittgenstein's point that "series" is ambiguous between
an extension, and a law (an ambiguity that lurks in Zenkin's definition,
when he talks of "the common *series* of the common finite natural
numbers" - what, all of them?).


AZ: 
The Wittgenstein's point is really very subtle. But the ambiguity lurks
not only 'between an extension, and a law' in definition of the series
(*) where the 'extension' is a priori understood as the potentiality and
the 'law' - as the actuality of the series (*). Indeed, the same Peano's
axiom "if n is natural then n+1 is natural too" defines well the
extension of (*) and simultaneously the law generating any n-th member
of (*), i.e., tells nothing about a true nature of the symbol '. . .'
(three dots) in the series (*). 
	The ambiguity and misunderstanding lurks also in the following.
	From childhood we are habituated to believe that if we have a
constructive algorithm (a law) allowing to construct ANY n-th member of
a well-ordered sequence then we have the sequence as a whole, i.e., we
have ALL members of the sequence and the sequence itself is considered
as actual. But the example of the series (*) shows that it is not so: we
have here the constructive algorithm (a law) 'n -- > n+1' allowing to
construct ANY n-th member of (*) but the algorithm (and the law) allows
us to state only that the series (*) has not a last element (i.e., it is
infinite), but it tells us nothing whether this infinity is potential or
actual.
	In a word, Aristotle defined the potential infinity, but he did
not PROVE that the infinite series (*) is really potential; Cantor
defined the actual infinity, but he did not PROVE that the infinite
series (*) is really actual. It is impossible to prove/disprove the
property of an infinite sequence (set) 'to be actual'; it is possible
only to accept/reject this property, i.e., (in general) the property of
the infinity 'to be actual/potential' has an AXIOMATIC character, and we
have here a situation which is completely similar to the history with
the V Postulate on parallels in Euclidian axiomatic. We have here the
following two opposed axiomatic statements.

	ARISTOTLE'S AXIOM. All infinite sets are potential.
	CANTOR'S AXIOM. All sets are actual (since all finite sets are
actual by definition).

	And consequently we have today in reality two very different
mathematics of infinity: the classical (really working) mathematics
based upon the Aristotle's axiom, and the mathematics of transfinite
'numbers' (set theory) based upon the Cantor's axiom. And either axioms
must be formulated explicitly in order to avoid in future, at last, the
vague discussions as to whether actual infinite exists or does not.
	However, there is a quite strange objection among some set
theorists and symbolic logicians: what for to formulate the Cantor's
axiom in the explicit form if all set theorists know well and
unanimously consent to that all sets (of modern axiomatic set theory)
are actual? I believe that behind this objection the natural intuitive
fear stands: they feel that such the explicit formulation of the
Cantor's actualization of infinite sets can lead to some undesirable
results 'in the light of (Aristotle's) logic'. Other logic or
mathematical grounds for such the strange objection are simply absent,
since, for example, an explicit formulation of NECESSARY conditions of
proofs of mathematical theorems does not contradict to any laws of logic
and mathematics, but moreover such the formulation is necessary for the
logical legitimacy of mathematical proofs.

DB:
W. imagines a machine that makes coiled springs, in which a wire is
pushed through a helically shaped tube to make as many coiled springs as
needed. What is called an infinite helix need not be anything like a
finite piece of
wire: it is the law of the helix.  Hence the expressions "infinite
helix" and "infinite series" are misleading for the same reason.
Nowadays, the analogue for the helical tube that is the matrix for an
infinite long spring, is the bit of source code that loops to produce an
"unending" sequence of integers.

AZ: 
Sorry, but it is too vague and free interpretation of W's 'machine' in
order to be discussed on a non-vague level.

DB:
Wittgenstein's objections (in the final sections of Philosophical
Grammar) are neither gross nor obvious, more of which later.

AZ: 
Sorry, but there is another Wittgenstein's objections (see W.Hodges, An
Editor Recalls Some Hopeless Papers. - The Bulletin of Symbolic Logic,
Volume 4, Number 1, March 1998. Pp. 1-17.) which is quite "gross and
obvious": "Cantor's argument has no deductive content at all", i.e.,
according to Wittgenstein, the famous, basic, and crucial for all modern
set theory Cantor's diagonal proof of the uncountability of real numbers
has no relation to what is called a deductive inference in classical
logic and mathematics.
	A very natural re-construction and logical continuation of this
W's objection (with quite unexpected consequences) is described in the
paper "On one re-construction of Wittgenstein's objection against
Cantor's Diagonal Method". - VII Scientific Conference "Modern Logic:
Problems of Theory, History, and Applications in Science",
Sankt-Petersburg, Russia, 20-22 June, 2002. Conference Proceedings, pp.
320-323 (in Russian). A short English version of the paper can be
emailed to those who is interested in the problem.

DB:

Dean

Dean Buckner
London
ENGLAND

Work 020 7676 1750
Home 020 8788 4273
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AZ:
	Alexander Zenkin

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Prof. Alexander A. Zenkin
Doctor of Physical and Mathematical Sciences
Leading Research Scientist
Department of Artificial Intelligence Problems
Computing Center of the Russian Academy of Sciences
Vavilov st. 40, 
117967 Moscow GSP-1, 
Russia

e-mail: alexzen at com2com.ru
URL:  http://www.com2com.ru/alexzen/ 
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^