Re: Continuum hypothesis



george wrote:

HZ and everybody else do in fact
have a particular model or structure (as you called it) in mind,
although here it is IMPORTANT to call it a model as OPPOSED to a
structure because the ONLY structures that are admissible to this
context (of differing truth-values) are ones that ARE models of the
governing AXIOMS.

Perhaps there is confusion as to just what axioms we are modeling
OVER.

Bingo.

We have been talking about 2nd and 3rd order arithmetic.
But we have also been talking about ZFC.

That probably explains my head twisting round and round.

2nd-order arithmetic has 1 language and 1 set of axioms.
ZFC has another.
I presumed we were going to be "compiling" the language of arithmetic
into ZFC and that models of ZFC (including its "standard" one, which
IS THE ONE that Rupert is talking about when he talks about "truth")
were going to be relevant.
But at 2nd-order, the Peano axioms become categorical, so it gets a
little harder to see where the shifting truth-values could be coming from.

Right.

Cohen's independence results are for 1st-order ZF.

See, I'm completely dead to the relevance of this remark.

To add to the confusion:

It seems that second order logic is not complete ... disturbing.
I presume this is also true of higher order logics.

It is also ironic: at second-order, PA and the theory of complete
ordered fiels are categorical. If they're not complete theories,
I guess it's because the logic is not complete, rather than because
there are non-isomorphic models.

--
hz
.

Relevant Pages