[FOM] Continuum Hypothesis

[Harvey Friedman]

Reply to Mangin 5/16/03 1:33PM and Mathias 5/16/03 10:45AM.

Mangin writes:
>
>1) Do you believe that the continuum hypothesis is true, or false?

I personally do not think about the continuum hypothesis in these 
terms. I think about

*what kind of results would shed light on the status of the continuum 
hypothesis?*

Here "status" means truth, falsity, and more broadly, notions of 
meaningfulness and meaninglessness.

I have not succeeded in doing anything important on this, although I 
do have some ideas in this direction that I would like to pursue. 
They are just ideas.

>2) Is there any general consensus amongst the mathematical/FOM 
>community regarding the truth or falsity of CH?

The mathematical community obviously has no general consensus, as 
they do not think about it. And they like the fact that they do not 
feel compelled to think about it.

There is no consensus among the f.o.m. community, the logic 
community, or even the set theory community regarding the truth or 
falsity of CH. In fact, there is not even a consensus among any of 
these communities as to the status of CH, which includes various 
kinds of meaningfulness or meaninglessness.

>3) What are the most important recent developments post-Cohen which 
>have contributed to this consensus (or lack thereof)?

The lack of consensus as to meaningfulness has been fed by the fact that

i) on the one hand, CH demonstrably holds in L, and in the analog of 
L where large cardinals abound (inner models), and

ii) every (countable) model of every reasonable large cardinal 
hypothesis has plenty of forcing extensions in which CH fails.

For some but not all, this casts doubt on the meaningfulness of CH.

Also, the lack of a compelling thematically striking general 
principle about the nature of sets after such a long time, for some 
but not all, casts doubt on the meaningfulness of CH.

Feferman has been most vocal among famous logicians that "CH is not 
meaningful".

>Have set-theorists proposed any plausible axioms which might decide 
>CH? Have any consequences of CH been discovered which either 
>strongly support or strongly undermine it?

There is no consensus among logicians, and apparently not even among 
set theorists, that anyone has presented reasonably compelling axioms 
that decide CH, or might decide CH. Nor consensus about any 
consequences of CH which either strongly support or strongly 
undermine it.

The two most visible developments in this regard are a very 
elementary argument of Freiling, which can be readily understood by 
virtually all logicians, and a great many mathematicians, and work of 
Woodin which cannot be readily understood by any but a very few 
experts in set theory.

The Freiling approach has been roundly rejected as grossly inadequate 
by the set theory community, and even based on a misunderstanding of 
the nature of set theory. There is a real consensus for you!

The Woodin approach is regarded as subtle, involved, and technical by 
the consensus of set theorists. There is hope among many in the set 
theory community that it will eventually be put in a form that would 
count as "compelling and thematically striking". However, there is 
also a consensus in the set theory community that this is appearing 
more and more to look like an inappropriate standard for work on CH.

Mathias writes:

>A good start would be the two articles by W. Hugh Woodin in the
>Notices of the American Mathematical Society; the first was in
>the June/July issue, 2001, (Volume 48 number 6) pp567--576; the second in
>the August issue of that year (Volume 48 number 7) pp 681-690.

I hesitate to call that a "start".

Rather than send readers to these papers, I am sure that the FOM 
would greatly appreciate Mathias' viewpoint on the Woodin approach, 
boiled down to a form in which the readership can understand. What 
does Mathias think of it?

>
>A lot has happened since Cohen. In the 60s it seemed that there would be
>little to choose between different possible values, amongst the alephs, of
>the continuum; but in the light of work starting in the 80s, it seems that
>the three contexts that permit the richest structure to the continuum are
>
>1) the continuum well-ordered and of size aleph_1
>
>2) the continuum well-ordered and of size aleph_2
>
>3) some version of the axiom of determinacy true and the continuum
>therefore not well-orderable.
>

L, and the analogs of L with lots of large cardinals (inner models), 
give rise to 1).

The Woodin approach "argues" for 2).

3) is totally different, and is incompatible with ZFC.

What is the relationship between "permitting the richest structure" 
and "the truth?"