[FOM] CH in standard models

[Roger Bishop Jones]

On Saturday 24 May 2003  9:25 pm, Harvey Friedman wrote:
> >In the responses to Sephorah Mangin it appears that:
> >
> >(1) there is no consensus about whether CH is true or false
> >(2) there is no consensus about whether CH is meaningful
>
> I agree.
>
> >It seems plausible that some of the conflicting evidence
> >and argument relating to (1) derives from the lack of clarity
> >about the meaning of CH which is evident from (2).
> >
> >I would guess that if the questions are sharpened to
> >speak specifically of the truth if CH in standard models
> >of (say) ZFC, then the situation is somewhat better.
> >(where "standard" is to be understood by analogy with
> >standard models of second order logic, i.e. the standard
> >models are the models of second order ZFC, V(alpha) for
> >alpha strongly inaccessible)
>
> I don't quite know what definition you want of "standard
> model". Here are some possibilities.
>
> A. The model of ZFC is an initial segment of the cumulative
> hierarchy of sets under epislon.
>
> B. The model of ZFC is a model of second order ZFC.
>
> C. The model of ZFC is a transitive set under membership.

I did say "standard models of second order ZFC".
I cannot see any difference between your A and B
(unless the omission of "standard" from B is to be read
as allowing non-standard models).
Aren't both A and B the V(alpha) for alpha strongly inaccessible?

C on the other hand seems to exclude no models, and
so is not what I intended.

> 1. The CH either holds in all A models or fails in all A
> models. If there is an A model then the CH holds in it if and
> only if the CH is true.
>
> 2. The CH either holds in all B models or fails in all B
> models. If there is a B model then the CH holds in it if and
> only if the CH is true.
>
> 3. If there is a C model then the CH model holds in some C
> models and not in other C models.
>
> All three statements 1,2,3 are provable in a weak fragment of
> ZFC without the power set axiom.

So I take it that you are agreeing that with a suitable
understanding of what "standard model of ZFC" means, there
would be a consensus that the question of whether CH is true
in standard models is meaningful?

This is all setting the stage for the question which
was the point of my message, viz:

If consideration is given specifically to the problem
of the truth value of CH in standard models then at least
some of the conflicting evidence related to the more
general question becomes irrelevant.
For example, the truth of CH in L or in any model obtained
by forcing is perhaps not relevant to the more specific
question, since these models are not standard.

I wonder if anyone could say more about how much
of the conflicting evidence for and against CH
falls by the wayside if the more specific question
of its truth in standard models is considered?
Most especially whether Woodin's considerations against
CH apply to this case.

Do you have anything to say on this?

Roger Jones