Re: Aleph One Sets

From: Tim Peters (tim.one_at_comcast.net)
Date: 11/03/04 

Date: Tue, 2 Nov 2004 21:50:22 -0500

[Mike Oliver]
>> Well, of course CH might turn out to be true,
>> even if we don't assume it.

[George Greene]
> Wrong. Jesus.

Sorry, I missed what Jesus has to do with this <wink>. Outraged posturing
is unattractive all the same.

> If ZFC is even meaningful (i.e. consistent),
> if there are any models of ZFC at all, then there
> are models of ZFC in which CH is false.
> So what can you possibly even MEAN by
> "CH might be true"??

Mike can speak for himself, but I know what it means to me: that we may
eventually discover that the consequences of ~CH are so contrary to what we
want from set theory that models wherein ~CH is true become uninteresting
except to set theory specialists. Or we may find that the consequences of
CH are unacceptable. Or neither, or both. Or we may find a compelling
axiom that implies CH, or ~CH. Everyone in the field, from Cantor to
Woodin, has speculated about whether CH "is true" in this sense. Most after
Cantor seemed to think "it's false" (including Godel and Cohen, who was the
first person to whom the independence of CH wasn't news <wink>).

As Mike said, Woodin's work is mentioned most often today in connection with
CH, and is often interpreted as casting strong doubt on "let's assume CH"
remaining a mainstream choice. Woodin himself is pretty non-committal; at
the end of "The Continuum Hypothesis, Part II" (Notices of the AMS, Vol 48,
No 7) he says:

    So, is the Continuum Hypothesis solvable? Perhaps I am not
    completely confident the "solution" I have sketched is the
    solution, but it is for me convincing evidence that there _is_
    a solution. Thus, I now believe the Continuum Hypothesis is
    solvable, which is a fundamental change in my view of set theory.
    While most would agree that a clear resolution of the Continuum
    Hypothesis would be a remarkable event, it seems relatively few
    believe that such a resolution will ever happen.

    Of course, for the dedicated skeptic there is always the "widget
    possibility". This is the future where it is discovered that
    instead of sets we should be studying widgets. Further, it is
    realized that the axioms for widgets are obvious and, moreover,
    that these axioms resolve the Continuum Hypothesis (and everything
    else). F or the eternal skeptic, these widgets are the integers (and
    the Continuum Hypothesis is resolved as being meaningless).

    ...

    The universe of sets is a large place. We have just barely begun
    to understand it.

> Even AFTER you think you have stumbled upon whatever
> allegedly conclusive evidence might "force" you to conclude
> that CH is true, The models in which it is false are NOT going
> to STOP existing!

Models of ZF in which the axiom of choice is false didn't stop existing
either, although most working mathematicians are happy to ignore them
completely -- AC is too useful to give up, and its odd consequences (like
Banach-Tarski, and that all sets can be well-ordered) aren't disturbing
enough to dissuade most workers from embracing ZFC wholeheartedly. If you
ask me what I mean by sets, then AC is "obviously true", and models of ZF in
which AC is false simply aren't models of what I mean by sets. It remains
possible that CH will meet a similar fate.

> ...
> No, if it's true, it will HAVE to be because you assumed it.
> It will HAVE to be because you assumed that, among the MANY
> available models of ZFC, one of the ones in which it
> HAPPENS to be true is THE "right" one.

It would be enough that ~CH turns out to have consequences few will accept,
so that CH must be true in all models most will accept. Or vice versa. You
won't get a categorical model of ZFC by adding CH or ~CH either, so "THE
'right' one" wouldn't necessarily apply.

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