Re: Implementable Set Theory and Consistency of ZFC

On Oct 23, 1:18 am, Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx> wrote:

Proof: trivial.

My remarks to follow are not meant to pertain to whatever "proof" is
supposed to be going on here, but rather to the points themselves:

In the beginning, there was no infinity.

In the beginning of what? And what does it matter what was the case in
whatever "beginning" you're talking about?

The first four
axioms of ZFC are sufficient for building up a set theory.

So what? We can restrict to even less than the four axioms or expand
to more than the four axioms, to get more models or fewer models,
respectively, of a theory in a language whose only non-logical symbol
is a 2-place relation symbol. We can even get a certain amount of work
done with no non-logical axioms at all, for that matter (cf. Quine's
'Set Theory And Its Logic'). Your preening about having discovered for
yourself (we've known it from the git-go) that a certain amount of set
theory can be done with fewer axioms is pathetic.

The next five
axioms are essentially redundant in this terse set theory.

Whatever "essentially redundant" means, separation (which makes the
axiom of the empty set redundant), power set, and infinity are used to
derive ordinary mathematics. Choice (or at least countable choice) is
also used for mathematics. Replacement (in suitable formulation) makes
separation redundant and is not ordinarily needed for mathematics
other than for set theoretic concerns themselves, as is the case with
regularity too.

MoeBlee

.

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