[FOM] CH in standard models

John Steel steel at math.berkeley.edu
Sun May 25 20:06:14 EDT 2003

On Sun, 25 May 2003, Roger Bishop Jones wrote:

 > 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?

As Harvey pointed out, the question as to whether CH is true
in V_alpha's is just the question whether it is true, period.
This is the question (some) set theorists are interested in, so
the suggestion that they focus on it is superfluous. Information
as to what holds in various models of set theory can, some think,
help attack this question.

John Steel

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