FOM: Proper Names and the Diagonal Proof

[Neil Tennant]

Correction to my last posting:

The version of Cantor's theorem that I gave has as an immediate corollary
what Boolos calls Not 1-1, but not (as I over-hastily claimed) what he
calls Not Onto.

On re-reading his note, I was struck by what seems to me a much-too-strong
demand that Boolos makes for a proof of Not 1-1 (i.e. no f maps P(X) 1-1
into X) to be "constructive". For any given f, he exhibits distinct
subsets A and B of X such that f(A)=f(B). But for the constructivist, this
is actually overkill, is it not? All the constructivist really has to do
is provide a reductio of the pair of assumptions

	for every subset Y of X, f(Y) is in X;
	for all subsets Y, Z of X, if f(Y)=f(Z) then Y=Z.

Such a reductio need not provide such subsets A and B as Boolos does.

___________________________________________________________________
Neil W. Tennant
Professor of Philosophy and Adjunct Professor of Cognitive Science

http://www.cohums.ohio-state.edu/philo/people/tennant.html

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