FOM: Cantor's Theorem & Paradoxes & Continuum Hypothesis

[Joseph Vidal-Rosset]

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.
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Joseph Vidal-Rosset
page web: http://www.u-bourgogne.fr/PHILO/joseph.vidal-rosset