FOM: Cantor's Theorem & Paradoxes & Continuum Hypothesis

[Neil Tennant]

On Tue, 13 Feb 2001, 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).

I don't see any threat of impredicativity in the use made of Separation in
the proof of Cantor's theorem. The "diagonal" subset D of X that is
obtained by Separation is defined as

	{x in X | x not in f(x) }

where f is assumed to be defined on X, 1-1, and taking subsets of X as
values. But in the condition "x not in f(x)" there is no *quantification
over* any totality of which D is a member.

The general misgiving about impredicativity and Separation is surely to do
with the fact that in certain instances of the Separation axiom scheme

	if the set X exists then so does the set {x in X | F(x)}

there will be instances phi (substituends) of the schematic letter F that
contain quantifications that *do* range over totalities to which the set
{x in X | phi(x)} belongs.

But, for the particular formula "x not in f(x)" used for the application
of Separation in the proof of Cantor's theorem, this is not the case.

Neil Tennant