[FOM] The Continuum Hypothesis

[Harvey Friedman]

Reply to Bagaria 11:39PM 5/18/03.

>Reply to Sephorah Mangin:
>
>>2) Is there any general consensus amongst the mathematical/FOM community
>regarding the truth or falsity of CH?
>
>I believe that some recent results on the continuum problem, most of them by
>Woodin, may lead in the next few years to a more or less general consensus
>among set theorists that the most reasonable cardinality for the continuum
>is aleph_2.

Do you think that there is a consensus among set theorists that the 
CH has a definite truth value?

Are you saying that among set theoretic statements being proposed for 
study that fix the value of 2^omega, the most interesting ones to 
study fix it to be 2^omega?

>
>>3) What are the most important recent developments post-Cohen which have
>contributed to this consensus (or lack thereof)?
>
>The most important contributions are due to Woodin. Besides the two survey
>papers by Woodin referenced in Mathias' yesterday posting, I would like to
>recommend Patrick Dehornoy's survey paper on Woodin's work on the Continuum
>Hypothesis, recently presented at the Seminaire Bourbaki. It is available,
>both in French and English, at Patrick's home page
>http://www.math.unicaen.fr/~dehornoy

When complicated references are given for subscribers to look at, I 
often ask subscribers (the author or others) to give their own 
account of and viewpoint on the main points. So I would like to 
request this here.

Regarding the work that has been done since the early results (that 
you can find plenty of forcing extensions with CH true and CH false, 
and CH holds in inner models): do you think it is significant from 
the point of view of f.o.m. and philosophy of mathematics? If so, in 
what way?

>
>>Have set-theorists proposed any plausible axioms which might decide CH?
>
>Yes. One example is Woodin's (*) axiom. Another is the Bounded Martin's
>Maximum (BMM). Both imply that the size of the continuum is aleph_2.

It would be useful if you or another subscriber could more or less 
state these two principles, and indicate why you think that they are 
plausible.