[FOM] Thoughts on CH 2

[Harvey Friedman]

It is my impression that just about all of the current approaches to CH
have at their basis, the following flawed "proof" of not CH.

Let M be a countable transitive model of ZFC. Then we can extend M in the
usual way by forcing to a countable transitive model of ZFC M' with the
same ordinals such that CH is false.

In particular, if CH holds in M, then we can add omega_2 mutually Cohen
reals and get c = omega_2. This is already due to Cohen.

So, the "argument" goes, if we want to MAXIMIZE then we must essentially
have omega_2 reals and not CH.

There are a number of flaws in this argument, not the least of which is
that "yes, but we can then make another generic extension, adding no new
reals, but collapsing omega_2 to omega_1, thereby restoring CH".

There are other flaws in this argument.

1. Whereas countable transitive models are obviously "extrinsically"
important, they are not at least prima facie "intrinsically" fundamental.
On the face of it, having a countable universe of sets must be only some
sort of technical device or perhaps some sort of approximation to reality.

2. Whereas forcing is obviously "extrinsically" important, it is not at
least prima facie "intrinsically" fundamental. It is, prima facie, a
technical device.

But the biggest problem that needs to be first addressed is that we can
extend to violate CH, and then extend again to restore CH.

The natural move is of course to declare one of these kinds of extensions
illegal. I.e., the second kind. But with what rationale?

An attempted rationale would be the following. Since equinumerosity is to
fundamental to set theory, we only want to consider extensions that
preserve equinumerosity. I.e., two sets in a model are required to be
equinumerous in any legal extension if and only if they are equinumerous in
the ground model.

Now the argument goes like this. Since every countable transitive model has
a legal extension with not CH, and every legal extension of every countable
transitive model with not CH must have not CH, THEREFORE CH is true
according to MAXIMIZE.

So we are using a principle which can be informally stated as follows.

DEFINITION. A universe is a model of ZFC. One universe strongly extends
another if and only if it extends it in the usual sense and also preserves
equinumerosity.

PRINCIPLE A. Let phi be a sentence. Suppose every universe has an imaginary
strong extension satisfying phi. Furthermore, suppose any strong extension
of any universe satisfying phi must satisfy phi. Then phi is true in this
universe.

I am under the impression that this sort of fairly simplistic thing has
been intensively investigated treating "imaginary universes"  in terms of
set forcing and class forcing, and "strong extensions" in terms of ccc
forcing extensions, or more generally cardinal preserving set and class
forcing extensions. I can see that perhaps some phi chosen to be connected
with "Jensen coding up the universe by a real" causes real problems with
this approach.  THere are obvious amalgamation issues.

Assuming that this kind of idea has been dead and buried, I would like to
offer some sort of possible partial resurrection as follows.

Firstly, enumerate a practical FINITE catalog of important, natural, and
famous set theoretic conjectures phi_1,...,phi_k, which of course icludes
CH.

Now state Principle A ONLY for phi_1,...,phi_k. What happens? Maybe in
practice (in this sense) we can make use of Principle A? Maybe if we
restrict to phi of a certain interesting kind (which would include
uninteresting cases as usual), Principle A can be salvaged?

If Principle A can be made to work even in this limited way, using
commonplace technical formulations, then I think that issues 1,2 above can
be satisfactorily handled foundationally.

Now for another unrelated point. Consider

*) there is a countably additive probability measure on all sets of reals
where the measure of points is 0.

This is well known to imply not CH. Furthermore, it is directly related to
a mathematically interesting and well known concept - countably additive
probability measures. I would assume that "all" set theorists reject this
solution to CH. A challenge for CH research is to explain why proposals
like *) and perhaps a workable form of Principle A above are far less
satisfactory than current CH research.

The usual answer I think is that there is no detailed structure theory
associated with *). But maybe something interesting structurally can be
done with *).

Harvey Friedman
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