[FOM] Cantor's argument
Alexander Zenkin
alexzen at com2com.ru
Sat Feb 1 08:48:42 EST 2003
The Wilfrid Hodges' article as a whole is fine, but there is one
doubtful point in it. Viz W.Hodges writes:
"The commonest manifestation <AZ: against Cantor's argument> was
to claim that Cantor had chosen the wrong enumeration of the positive
integers:
x1, x2, x3, . . . (C)
His argument only works because the positive integers are listed in such
a way that each integer has just finitely many predecessors. If he had
re-ordered them so that some of them come after infinitely many others,
then he would have been able to use these late comers to enumerate some
more reals, for example the <AZ: Cantor's anti-diagonal> real number
<AZ: x* which is different from each real in (C)>."
W.Hodges believes that only such indexing of reals in (C) are
admissible which use ALL natural numbers of the set N={1,2,3, . . .},
since only such indexing allows "to reach Cantor's conclusion", but any
re-indexings of the reals in (C) which use NOT ALL natural numbers are
inadmissible since they lead to conclusions contradicting to Cantor's
one. Unfortunately, he did not formulate a mathematical or logical
criterion to make such a differentiation.
Remind the Cantor's argument. Let X=[0,1].
Cantor's THEOREM. X is uncountable.
Cantor's PROOF (by Reductio ad Absurdum, further - RAA).
Assume that X is countable.
Then there is an enumeration (C) of ALL reals of the set X.
Applying to the list (C) his famous diagonal method, Cantor
produces a NEW real number, say x*, which differs from every real in
(C). Consequently, x* does not belong to (C), i.e. the enumeration (C)
does NOT contain ALL reals of the set X. - Contradiction. Q.E.D.
However, from the mathematical point of view, the assumption "X
is countable" means that X is equivalent to ANY other countable set,
i.e., there is a 1-1-correspondence between reals of X and elements of
ANY countable set, e.g., N={1,2,3,4,5,. . .}, N1={1,3,5,7,9,. . .},
N2={2,4,6,8,10,. . .}, N3={2,3,5,7,11,13,. . .} and so on up to
infinity. It is obvious that if we re-index all reals in (C), - not
changing their number and order (!), - with elements, say, of N1 then
we shall always be able to index any (even infinite!) set of NEW
Cantor's anti-diagonal reals with elements of the set, say, N2 and the
final Cantor's statement that the set X has much more elements than the
set N will become very not-evident.
So, the only reason to choice just the set N of ALL natural
numbers to index reals in (C) and to forbid all other, mathematically
admissible and absolutely equal in logical rights, alternative,
countable sets N1, N2, N3, etc., is a hot DESIRE "to reach Cantor's
conclusion" that X is uncountable. But the choice of one alternative
(which leads to Cantor's conclusion) from an infinite set of others
(which lead to conclusions contradicting to the Cantor's conclusion)
according to one's wish is, according to Kronecker, a teleological
activity, but not a mathematical one.
So, I state that the known final W.Hodges' conclusion ". . .
there is nothing wrong with Cantor's argument" is not correct.
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/
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-----Original Message-----
From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf
Of Jim Farrugia
Sent: Friday, January 31, 2003 11:17 PM
To: Alasdair Urquhart
Cc: fom at cs.nyu.edu
Subject: Re: [FOM] Cantor's argument
There is an article by Wilfrid Hodges in the Bulletin of Symbolic Logic
(Vol 4. No. 1 March 1998) called "An Editor Recalls Some Hopeless
Papers"
that might be of interest to those involved in this thread.
A URL with a link to the article is
http://www.math.ucla.edu/%7Easl/bsl/0401-toc.htm
Jim
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