[FOM] Continuum Hypothesis

[Paul Larson]

>1) Do you believe that the continuum hypothesis is true, or false?
>2) Is there any general consensus amongst the mathematical/FOM community
>regarding the truth or falsity of CH?
>3) What are the most important recent developments post-Cohen which have
>contributed to this consensus (or lack thereof)? 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?


I prefer to think of this issue in terms of: in the project of formalizing
the notion of the universe of mathematical objects, should we include CH,
its negation or neither? Even under this reformulation, many people seem
to believe that the answer is definitively neither, either because the
question is vague, or because ZFC doesn't decide the answer (other people
believe we shouldn't be pursuing this project, or that we shouldn't be
pursuing it in terms of sets, or that we shouldn't use Zermelo-Fraenkel
set theory, etc).  My point of view, which I believe is held by many
current set theorists, is that the answer is neither, for now, but that
this might change. I will confess that I do not know of a sentence whose
proof would convince me of either truth value of CH, though a proof of the
following (paraphrased) conjecture of Woodin on the first page of the
paper by Dehornoy which Joan mentioned (incidentally, I have found that
some computers can access Dehornoy's page and some can't) would strike me
as strong evidence: there is an extension of ZFC compatible with all large
cardinals which fixes the theory of the powerset of omega_1 with respect
to forcing, and every such theory implies that CH is false.  Woodin's
axiom (*), which I define below, gives an extension which fixes this
theory, but has not been shown to be consistent with large cardinals. If
Woodin's Omega conjecture (also described below) is true, then (*) is
consistent with all large cardinals.

I think it is helpful to consider the analagous question for the following
statements:

- statements of finite combinatorics which are equivalent to the
consistency of certain large cardinals
- V = L
- Projective Determinacy

Accepting truth values for any of these statements, which many set
theorists do, would seem to rule out the "ZFC does not decide it" argument
above in the case of CH. However, it is my impression that there is not
even a consensus on the truth values of statements of the first type
listed above.

The argument by vagueness might say that statements about the integers
have truth values, but at some point, as we move to statements about
arbitrary real numbers, sets of real numbers, etc. we gradually (or, at
some point, sharply) lose contact with reality. This picture makes some
intuitive sense, but I don't think I've seen a convincing defense of it.
Perspectives change as we learn more mathematically. On the other hand,
there remains the possbility that independence results for the integers
will bring this vagueness to their level.

Many set theorists now believe that PD, or the stronger statement that
Determinacy holds in the inner model L(R), should be accepted. This point
of view is often suported by the claim that PD gives a deep and natural
theory for the reals extending the classical theory, and the fact that PD
follows from every natural (i.e., ruling out self-referential statements
and statements of consistency) mathematical statement of sufficient
consistency strength.

Even if we accept answers for the statements listed above, the argument
still remains against CH that, unlike the truth values of the statements
listed above, its truth value can be changed (over any model) by forcing.

This is where Woodin's work comes in. A quick summary of some of it (a
long list of caveats supressed):

Omega-logc. In first order logic, any model satisfying a given theory
witnesses the consistency of that theory. The idea in Omega-logic is that
not just any model is good enough. Making this precise uses the notion of
the universally Baire sets of reals. These sets have definability
properties which give them unambiguous interpretations in forcing
extensions. The natural ways of formalizing this idea are equivalent in
the presence of a proper class of Woodin cardinals (the ambient theory for
the results in this paragraph). Given a universally Baire set of reals A,
a model M is A-closed if the intersection of M and A is in M, and whenever
we force over V with a partial order in M, the corresponding extension of
M interprets A correctly. When A is the complete Pi^1_1 set, this just
means that M is wellfounded, and as A gets more complex this is a stronger
and stronger requirement on M. Now, a theory T is Omega-consistent if for
every universally Baire set of reals A, T holds in an A-closed model.
Suitable initial segments of V and of its forcing extensions are A-closed
for all A, so any sentence which is provably forceable is
Omega-consistent. In particular, CH is not resolved in Omega-logic.
However, if the theory of the powerset of omega_1 is finitely
axiomatizable in Omega-logic, then CH fails, and moreover there is a
definable counterexample. Woodin's Omega Conjecture states (roughly) that
for Sigma_2 senteces (over V), consistency in Omega-logic is the same as
forceability.

Pmax. Woodin's partial order Pmax is a forcing construction in L(R).  If
L(R) satisfies the Axiom of Determinacy, the powerset of omega_1
(P(omega_1)) in the Pmax extension of L(R) is the direct limit the
P(omega_1)'s of a collection of models which includes cofinitely often
models satifying every Omega-consistent extension of ZFC. Pmax can
reasonably be seen then as producing the completion of P(omega_1).
Woodin's axiom (*) is the statement that AD holds in L(R) and that
L(P(omega_1)) is a Pmax extension of L(R). Axiom (*) also implies that
there is a definable counterexample to CH. Axiom (*) is Omega-consistent,
but it is open whether large cardinals imply that it can be forced over V
- in the usual case, it is forced over the inner model L(R).


Some other set theoretic arguments related to the truth value of CH:


At this point there are approximately eight separate arguments, from
various hypothesis, for the statement that the continuum is less than or
equal to aleph_2. The arguments tend to follow from consequences of
Martin's Maximum or Axiom (*), but the arguments themselves use distinct
consequences of these axioms.

Some people feel that CH is false because it simplifies the theory of the
reals in ways that they find counterintuive (for example enabling diagonal
arguments for creating pathological sets of reals).  Another argument
against it is that it wipes out the entire field of cardinals invariants,
the study of definable cardinals whose possible values are bounded by
aleph_1 and the continuum. A counterargument is that the continuum would
need to be much larger than aleph_2 to allow the full theory of cardinal
invariants to flower.  As a counter-counterargument, there exists a much
wider arrary of techniques for produing models in which the contiuum is
less than or equal to aleph_2 than for producing models in which this
fails. It may be that there are technical reasons for this which would
lend support to the conclusion that the continuum is less than or
equal to aleph_2.