FOM: Cantor's Theorem & Paradoxes & Continuum Hypothesis
Joseph Vidal-Rosset
jvrosset at club-internet.fr
Tue Feb 13 03:45:05 EST 2001
Dear Bill,
Thanks a lot for your reply and your help.
>Impredicativity, as I understand it, concerns definitions. A
>definition of M is impredicative if the the definition refers to a
>totality (e..g involves quantification over that totality) to which
>M belongs. I don't see that Cantor's argument is impredicative. It
>shows that, given a function F from a set A whose values are subsets
>of A, there is a subset C of A not in the range of F.The definition
>of C is: x in C iff x not in f(x). That is not an impredicative
>definition.
Yes, I recognize perfectly what I have learned about impredicativity.
>I also see no reference in this argument to the power set P(A) of A.
>P(A) gets mentioned only---and of necessity---in the conclusion that
>P(A) has power greater than A. If one doesn't like power sets, then
>just don't mention this result: Cantor's diagonal argument will
>nevertheless stand.
The point is that invoking Separation in the proof there is an
implicit mention of the Power Set: "a particular subset of A is
determined by the totality of all subsets of A, or even by the
totality of all sets - which is just the procedure against which
Russell's vicious circle principle was directed." (Fraenkel et alia,
Foundations of Set Theory, p. 38).
>As for whether impredicative definition is compatible with
>constructivity, note that at least Goedel thought that his theory of
>impredicative primitive recursive functions is constructive.
Yes, and I guess that the impredicative definition of "the most
typical Yale man" (famous Quine's example) is compatible also with
constructivity.
But I wonder if in Set Theory it is exactly the same story. Poincaré
had strong doubts also (because of the use of actual infinite). But
maybe these doubts are irrelevant, I really hesitate and I would be
very thankful to the FOM correspondent who would shed light upon this
question.
>Best wishes from Chicago, where the sun shone today!
Best wishes from Dijon, where the weather is hazy this morning, like
my ideas about this stuff.
Jo.
--
------
Joseph Vidal-Rosset
page web: http://www.u-bourgogne.fr/PHILO/joseph.vidal-rosset
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