FOM: Cantor's Theorem & Paradoxes & Continuum Hypothesis

Joseph Vidal-Rosset jvrosset at
Tue Feb 13 10:46:18 EST 2001

>At 9:45 AM +0100 2/13/01, Joseph Vidal-Rosset wrote:
>>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).

At 8:30 -0600 13/02/2001, William  Tait wrote:
>Jo, I don't see how this can be correct---pace Fraenkel. This would 
>mean that there is no such thing as predicative analysis, since, by 
>your criterion, *every* definition of a set of numbers, e.g. even 
>that of the set of even numbers, presupposes the totality of such 
>sets and so is impredicative.

Dear Bill,

Yes, I believe that is exactly the lesson of the complex story of 
impredicativity : roughly we need to introduce the set of integers 
and the definitions made on numbers are impredicative by the way. See 
for example the predicative  schema of systems constructed by Wang 
which shows perfectly this fact. Poincaré critics to Russell's theory 
of type can be made against Wang's system in the same way. Kreisel 
pointed out this fact very clearly. And it shows, interestingly from 
a philosophical point of view, in my opinion, the limits of 
constructivism, or the limits of predicativism, and that is a good 
point for the realism in mathematics, independently of the various 
meanings of this word.

At 8:30 -0600 13/02/2001, William  Tait wrote:
>Of course, in ZF, every definition of a set of objects requires 
>Separation. But suppose we dropped Separation (and even Power Set) 
>and introduced, as primitive, the rank function, rank(x). Then we 
>could have the axiom that any property of sets of rank <= alpha 
>defines a set. The rank function itself seems predicative: rank(x) = 
>union {rank(y) : y in x} would be its defining axiom.
>We shouldn't be bullied by set theory or some particular axiomatization of it.

Thanks a lot for this technical precision, that helps.

All the best,

Joseph Vidal-Rosset
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