[FOM] NEW SEMANTIC PARADOX GENERATED BY ACTUALIZATION OF INFINITY: WHETHER MATHEMATICS IS COGNIZABLE ?

[Alexander Zenkin]

Martin Davis wrote: 

>From: fom-admin at cs.nyu.edu 
>on behalf of Martin Davis [martin at eipye.com]
>Date: 4 January 2003, 20:58
>To: fom at cs.nyu.edu
>Subject: [FOM] FOM: set theory

>Dean Buckner wrote:

>>My view is that set theory is wrong,
>>in the sense it does not capture our 
>>ordinary intuitions about sets and things.

>This makes about as much sense as saying that physics
>is wrong because it doesn't capture 
>our ordinary intuitions about work and energy. 
>A scientific discipline is not intended to capture
>our ordinary (fallible) intuitions, but rather 
>to improve on them. Particularly on FOM, 
>set theory is of special interest in providing 
>a foundation for mathematics where "our ordinary intuitions" 
>have proved utterly misleading.
>
>Martin
>
>Martin Davis
>Visiting Scholar UC Berkeley
>Professor Emeritus, NYU
>martin at eipye.com (Add 1 and get 0)
>http://www.eipye.com 
>_______________________________________________
>FOM mailing list
>FOM at cs.nyu.edu
>http://www.cs.nyu.edu/mailman/listinfo/fom

	Further AZ = Alexander Zenkin for short.

	AZ:

	As H.Poincare said, "... logic and intuition play its own
necessary role. Both of them are unavoidable. Logic which only one can
give a reliability, is a tool to prove; intuition is a tool to invent."

	I agree with Martin Davis that set theory is intended to improve
on our ordinary (fallible) mathematical intuitions. But the example
below shows that there is a case where the set theory fulfils this
mission not too well.

	Recently Samuel S. Kutler wrote in his message to the [HM]-list
(HM=History of Mathematics).

>From: owner-historia-matematica at chasque.apc.org 
>on behalf of Samuel S. Kutler [reltuk at comcast.net]
>Date: 4 January 2003, 1:43
>To: historia-matematica at chasque.apc.org
>Subject: Re: [HM] existence statements in mathematics
>
>Friends:
>
>Mathematical existence is a very tricky subject!
>How many real numbers exist?
>Here is Willard Van Orman Quine on the subject
>page 273 of Mathematical Logic) Revised Edition,
>Harper Torchbook:

>   "The irrationals exist in such variety, indeed, 
>    that no notation whatever is capable of providing 
>    a separate name for each of them."
>
>Sam Kutler


	AZ: 

	What does the Willard Van Orman Quine's statement mean from
mathematical point of view?

	Let X=[0,1] be a set of real numbers of the segment [0,1].
	Our mathematical as well as meta-mathematical, set theoretical,
and even philosophical intuition prompts us that if we have shown two
points '0' and '1' then we have and can specify, individualize any point
x, 0<x<1. Whether the intuition is right?
	According to the most general Cantor's definition of the notion
of real number, a real number x (belonging to [0,1]) is an infinite
sequence of, say, binary digits:

	x = 0, x1 x2 x3  ... xn ... (A)

	To define a real number, x, (or really to choice x from X and to
differentiate it from all the rest of elements of X) in mathematical
sense means to give an algorithm which allows to construct any n-th
digit in the sequence (A). Thus if some real number has not such an
algorithm in principle (i.e., today and never) then such the real number
is an unknowable, i.e., non-existing, 'thing' in any (including
mathematical) sense.
	As is known, any (mathematical) algorithm is a finite string in
a finite alphabet and the set of all such strings is countable.
	If the infinite set X and the infinite sequences like (A) are
actual, then Cantor's theorem and its diagonal proof are valid and the
set X is uncountable.
	However, only a countable subset, say X1, of real numbers in X
will ever have the algorithms allowing to represent them in the form
(A), i.e., providing the reals with what we call 'a mathematical
existence'. It is obvious that the uncountable subset, say X2 (with
X1+X2=X), of real numbers will never have the algorithms allowing to
represent them in the form (A), and such the real numbers will never be
specified, shown, determined, existing, knowable, and cognizable in
mathematical sense.
	It is obvious that the ratio of the countable quantity of the
cognizable reals in X1 to the uncountable quantity of the incognizable,
unknowable reals in X2 is something like the ratio of '1' to 'oo', i.e.,
the ratio is (almost) zero. It means that we know in reality (almost)
nothing about the real numbers of the segment [0,1], i.e. about Cantor's
continuum. 
	The same refers to any function, f(x), 'defined' on the segment
[0,1], which is unknowable on the uncountable subset, X2, of reals. It
means that we know in reality (almost) nothing about any function given
on the segment [0,1] as well as on any other segment.
	The same refers to mathematical analysis as a whole, and we can
claim (together with Gregory Chaitin, at IBM's T.J.Watson Research
Center in Yorktown Heights, New York) that Cantor's continuum is "not
simply moth-eaten, it is mostly made of gaping holes", and mathematics
as a whole is simply an unknowable 'thing'. 
	Of course, iff the continuum is actual and uncountable in
Cantor's sense.

      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/ 
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