Re: Some basic set theory questions

David C. Ullrich <dullrich@xxxxxxxxxxx> writes:

No, the words "everything is a set" have not appeared in any of your
presentations of the proof. But the proof is valid only in
formalizations where the universe of discourse includes nothing but
sets.

I'll freely admit I haven't been following this exciting exchange in any
detail, but doesn't Moe's proof rather rely on a curious formulation of
separation, which allows us to form "subsets" of not only sets but of
any object at all, the formalisation as a scheme of

For any property P and an object A, there exists a set B such that
x in B iff x is in A and P holds of x.

?

This principle is of course rather peculiar, and certainly not something
we'd expect to find in any treatment of set theory conducted in
mathematical English -- it simply makes no ordinary mathematical sense
to ask whether e.g. pi is a member of the Diophantine equation 3x^2 + x
= 0 and so on.

T.H. Ray has shrewdly surmised I don't do any mathematics, so my
opinions probably count for very little. Be that as it may, I do find it
odd that people seem to feel very strongly that in discussions about set
theory and elementary proofs in set theory it is necessary to drag in
all sorts of formal grudgery of no apparent immediate relevance. In
contrast, a post asking how to prove that x^n * x^m = x^(n+m) would
probably not occasion an extended discussion about the restriction on
free variables in instances of the induction schema in first-order
arithmetic, the use of Gödel's beta-function to express exponentiation
in the language of arithmetic, and so on.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxx)

"Wovon mann nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.

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