# FOM: Cantor's Theorem & Paradoxes & Continuum Hypothesis

William Tait wwtx at midway.uchicago.edu
Tue Feb 13 09:30:46 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).

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.

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.

Best wishes,

Bill

```