[FOM] CH in standard models

[Harvey Friedman]

>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.

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.

Harvey Friedman