[FOM] CH in standard models

[Harvey Friedman]

Reply to Jones 5/25/03  6:20AM.

>
>  > 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?

No. The least cumulative hierarchy that satisfies ZFC is much shorter 
than the first strongly inaccessible cardinal (assuming it exists). I 
think this goes back to Vaught.

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

It includes exactly the well founded models.

>
>>  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?


No. The CH is outright equivalent to the question of whether the CH 
holds in all or some A models or B models (assuming they exist). So 
your use of standard models in connection with CH accomplishes 
nothing.

>
>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?


Since it is outright equivalent (in the sense stated above), 
absolutely nothing whatsoever falls by the wayside.