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

[Alexander Zenkin]

According to Hilbert, an infinite is not realized in the
physical, 'ordinary' world, and therefore our ordinary intuition is
based on a practice having to do only with finite things and sets.
Therefore we simply can't have an intuition of an infinite. Cantor's and
any modern axiomatic set theory have to do with just infinite sets, and
therefore they can't and must not capture our 'ordinal' intuition, by
definition. Cantor constantly accentuated that one should not transfer
properties of finite sets to infinite sets.
However he himself was not always keeping to this demand.
Indeed, e.g., the operation '+1' is strictly defined in Peano's
axiomatics for the finite natural numbers. But the application of the
operation '+1' to transfinite 'things' is not self-obvious (since finite
'things' and transfinite 'things' are very different entities in their
mathematical nature) and is in need of a strict ground. Cantor and
modern set theory did not demonstrate such the ground ever.
Nevertheless, using our (finite) intuition as to the obvious
Peano's axiom "if a 'thing' is integer, then the 'thing'+1 is integer as
well", Cantor simply replaces the intuitively obvious thing '0' in the
series
0+1, 0+2, 0+3, ..., 0+n, ... (*)
by a new NAME 'omega' (further - w) and 'constructs' the same series (*)
in a 'new form':
w+1, w+2, w+3, ..., w+n, ... (C1)
Then, on reaching the crucial NAME '...' in (*) he introduces a
new NAME, '2w', calls this NAME an integer and repeats the same
procedure:
2w+1, 2w+2, 2w+3, ..., 2w+n, ... (C2)
Gluing the strings (*), (C1), (C2), etc., in a one long string
and using a non-positional notations for (transfinite) 'integers',
Cantor generates his famous series of transfinite ordinal numbers up to
the ordinal 'epsilon-0', and goes on further in the same manner.
In such the case, the Cantor's operation '+1' means a
non-mathematical operation of a concatenation of strings (symbolic
NAMES), and his specific summing up, n+w=w, means a non-mathematical
operation of an absorption of some NAMES by other NAMES.
At that Cantor uses his ingenious invention, viz in order to
have an unrestricted source of new NAMES, he utilizes not common strings
in a finite alphabet, but different summands of a common algebraic
polynomial,
P(x) = . . .+at*x^t + . . . a2*x^2 + a1*x + a0,
as NEW NAMES to generate his transfinite ordinals "of the second kind"
(so-called 'limit ordinals'), where the coefficients ai are changeable
from 1 to w, the NAME of the variable x is replaced by the name w, and
an exponent t is also a polynomial written in a non-positional additive
notation with radix w as well as the initial polynomial P(w).
Such the Cantor's building does not contradict to our 'ordinal'
intuition, since it based (and does not overstep the frame of) the
second Peano's axiom: "if a 'thing' is integer then the 'thing'+1 is
integer as well" independently of what a deep philosophical sense (the
name 'w' is a completed, invariable series (*)) is included by us in the
term 'thing'.
By the way, the last fact proves strictly that Cantor's theory
of transfinite ordinals is CONDITIONALLY consistent: IFF the three first
Peano's axioms (giving the formal, inductive definition of the series
(*) ) are consistent.
Whether such Cantor's 'transfinite building' has a mathematical
sense is not too obvious.
Now some words as to a view "that set theory is wrong".
I agree but not because it does not capture our ordinary
intuitions about sets and things.
I agree that set theory (of transfinite cardinal 'numbers') is
wrong because this (Cantor's 'naive' as well as modern 'non-naive') set
theory is based on Cantor's theorem on the non-denumerability of
continuum the proof of which, in it's turn, is based on the conception
of actual infinite.
In my previous message "As to strict definitions of potential
and actual infinities" (see FOM-archive at:
http://www.cs.nyu.edu/pipermail/fom/2002-December/006121.html)
I have given a quite impressive list of Cantor's opponents as regards
the rejection of the actual infinite who, according to W.Hodges'
classification, "must be <AZ: considered as> ... dangerously unsound
minds" (see his famous paper "An Editor Recalls Some Hopeless Papers." -
The Bulletin of Symbolic Logic, Volume 4, Number 1, March 1998. Pp.
1-17, http://www.math.ucla.edu/~asl/bsl/0401-toc.htm ).
Now I would like to remind some of appropriate statements of
such the "dangerously unsound minds".
For example, Solomon Feferman writes (in his recent remarkable
book "In the light of logic. - Oxford University Press, 1998."):
"[...] there are still a number of thinkers on the subject (AZ:
on Cantor's transfinite ideas) who in continuation of Kronecker's
attack, object to the panoply of transfinite set theory in mathematics
[.] In particular, these opposing <AZ: anti-Cantorian> points of view
reject the assumption of the actual infinite (at least in its
non-denumarable forms) [...]Put in other terms: the actual infinite is
not required for the mathematics of the physical world."
The same view as to rejection of the actual infinite is clearly
expressed by
Ja.Peregrin (see his "Structure and meaning" at:
http://www.cuni.cz/~peregrin/HTMLTxt/str&mea.htm):
"There is not an actual infinity",
V.F.Turchin (see his "Infinity" at
http://pespmc1.vub.ac.be/infinity.html)),
"For actual infinity we have no place in [...] the global cybernetic
theory of evolution and in the constructivist foundation of
mathematics",
and many other modern experts in foundations of modern mathemnatics. So
we see that the acceptance of actual infinity is by no means unanimous
in contemporary mathematics. This is worth examining in greater detail.
Since so far no member of the FOM-list expressed any objection
as to my axiomatic and algorithmic definitions of the actual and
potential infinity (see the reference to the FOM-archive above), I
believe that the definitions are accurate from mathematical point of
view and will really help to make clear some doubtful and vague problems
of foundations of mathematics connected with the usage of actual
infinity in Cantor's set theory.
However, I don't completely agree with S.Feferman's cautious
rejection of "the assumption of the actual infinite (at least in its
non-denumarable forms)".
I believe that the property "to be actual" and "to be potential"
refers to the set property "to be infinite", but not to the number of
elements of the infinite set (i.e., to the infinite set cardinality).
Otherwise one must explicitly formulate a mathematical criterion
according to which the set of natural numbers with the cardinality
'aleph-0' is actual, but the set of real numbers (continuum) and the
sets of the cardinality, say, greater than 'aleph-0' are potential. I
think such the 'fine' differentiation can really transform the modern
set theory into the "luminiferous ether" which "is not detectable by any
means available to us", according to Mayberry (see "The Foundations of
Mathematics in the Theory of Sets", p. 267-8).
So, either all infinite sets are accepted 'to be potential' (as
Aristotle and modern, according to Feferman, 'really working'
mathematics do that), or all infinite sets are accepted 'to be actual'
(as Cantor and all modern set theories do that), but a third is not
given.

Further I believe that all the axiomatic set theories break down
the classical logic and the classical mathematics in the following
point.
In my previous message (see FOM-archive at:
http://www.cs.nyu.edu/pipermail/fom/2003-January/006137.html) I wrote:
"We have [...] 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 the
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? "
Following to such a 'logic', mathematicians must delete, say,
the axiom on parallels from Euclid's axiom system (since today every set
theorist knows well that two parallels don't intersect), or say, the
axiom 'if n then n+1' from Peano's axiom system (since today every set
theorist knows well how to add '1' to any given finite n). I suspect
that it would spawn a quite strange 'mathematics' and no mathematician
will ever agree to plunge its "Queen of all sciences" into such the
"luminiferous ether" of hidden axioms and concealed necessary conditions
of mathematical proofs which (these axioms and conditions) "are not
detectable by any means available to ..." any formal, axiomatic
theories.
So, the deliberate concealment of Cantor's axiom by modern
axiomatic set theory is a direct deception of mathematical community
and, paraphrasing Napoleon, "it is not an (logical) error, it is a
(mathematical and educational) crime" and, according to Brouwer, it is
"a pathological accident in history of mathematics from which future
generations will be horrified".
Some of a lot of quite unexpected logical consequences of the
explication of Cantor's axiom and new aspects of Cantor's diagonal proof
of the uncountability of continuum are described and analyzed in the
paper "SCIENTIFIC INTUITION OF GENII AGAINST MYTHO-'LOGIC' OF CANTOR'S
TRANSFINITE 'PARADISE'", presented at the International Symposium
"Philosophical Insights into Logic and Mathematics", Nancy, France,
30September-04October, 2002, and now acceptable at:
http://www.philosophy.ru/library/math/sci_intuition.pdf.
In the paper, it is shown, in particular, that the famous
Cantor's theorem on the uncountability of continuum was never comprising
at least two necessary conditions of its own proof in an explicit form.
The scandal, from the mathematical point of view, fact explains, in
particular, why nobody was able to disprove the theorem. The explication
of the conditions and their logical analysis prove that Cantor's
"diagonal argument" proves nothing: the first necessary condition makes
the "diagonal argument" invalid, the second necessary condition is
simply a teleological one having no relation to mathematics. Thus, the
notorious uncountability of continuum is not proved by Cantor, and our
results show that the main paradigms of foundations of modern
meta-mathematics (so-called 'proof theory') and 'non-naive' axiomatic
set theory as well as some basic deductive rules of the modern predicate
calculus must be essentially revised from the point of view of really
working, classical mathematics and classical Aristotle's logic.
Paraphrasing the known Harvey Friedman's quotation, I can now
ask: "Are you talking about set theory WITHOUT the Cantor's axiom? And
if WITHOUT, what are you talking about"?
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/
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^



                           Martin Davis
                    Visiting Scholar UC Berkeley
                      Professor Emeritus, NYU
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