[FOM] CH in standard models

Roger Bishop Jones rbj at rbjones.com
Sat May 24 11:14:36 EDT 2003

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

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)

In particular, I have the impression that:

(2') there is a consensus that the question whether CH
is true in standard models is meaningful (though not unanimity)

(since it seems generally accepted that CH has
the same truth value in all standard models)

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.

Roger Jones

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