[FOM] Continuum Hypothesis
steel at math.berkeley.edu
Mon May 19 02:47:22 EDT 2003
It is conceivable that some natural extension of our current
large cardinal hypotheses, some natural markers of consistency
strengths beyond the region we have explored, decides CH.
This would be a solution closest to the kind Godel hoped for.
Of course, such extended large cardinal hypotheses would have to
be significantly different from the ones we have now, and the ones
we have now go as far up the consistency strength hierarchy as
we seem to have any use for going now. So if there is a solution along
these lines, it isn't likely to emerge soon.
A second approach is based on the idea that preservation under small
forcing will be characteristic of all large cardinal axioms. We therefore
must augment the large cardinal axioms by, to speak colorfully,
saying which sort of generic extension we want to take as our
reference point. More precisely, we want to complete the large cardinal
axioms, and the metamathematical evidence of completeness is a
generic absoluteness theorem. The paradigm here is Woodin's
absoluteness theorem" for Sigma^2_1 sentences: granted sufficiently
strong large cardinal axioms, any two generic extensions of V which
satisfy CH satisfy all the same Sigma^2_1 sentences. Thus CH plus
large cardinals yields a "complete" (i.e. immune to independence by
forcing) theory at the Sigma^2_1 level.
To my mind, the best hope for a solution to CH in my lifetime
is the discovery of conditional generic absoluteness theorems
extending this one to all levels of the Sigma^m_n hierarchy, and beyond.
If this (and not the scenario in paragraph 1) is how things are,
then there might well be conditionally generically absolute theories
consistent with all large cardinal axioms, and "going all the way"
in the family of sentences they can decide, some of which have
CH and some of which have -CH. (The theories would be intertranslatable--
each could view the other as the theory of a certain kind of generic
extension of its own universe. So in this scenario, there is at some deep
level an ambiguity in the language of set theory.) It does seem more
promising now to look at theories containing CH, since we have Woodin's
Sigma^2_1 generic absoluteness theorem to build on then. A natural
question is whether diamond works at the Sigma^2_2 level the way
CH did at the Sigma^2_1 level.
Woodin's Omega conjecture has as a consequence that the conditional
generic absoluteness solution is impossible. It says that all
conditional generic absoluteness at the Sigma^m_n level is
enforced in a certain way by Hom-infinity sets, which a non-expert
can think of as meaning that it occurs for the reasons that
generic absoluteness occurs where we know it now. In Woodin's
language, every Omega^*-complete theory is actually Omega-complete.
(At the Sigma^m_n, or indeed Sigma_1-in-the-powerset, level.)
Woodin has shown:
Theorem (Woodin) No axiomatizable, Omega-consistent theory
is Omega-complete for all Sigma^2_3 sentences.
The Omega-conjecture is a very important open problem. If it is true,
conditional generic absoluteness (= Omega^*-completeness) is the same as
Omega-completeness, and thus by the theorem just quoted, is not an
appropriate criterion for evaluating theories in general, or even at
the Sigma^2_3 level.
Woodin goes on to argue that a proof of the Omega-conjecture would
constitute evidence for -CH, but I disagree here. His argument is that
there are Omega-consistent theories which are Omega-complete for the
theory of L(P(omega_1)) (such as the Pmax axiom saying that L(P(omega_1))
is a Pmax extension of L(R)), while by the theorem above, any such theory
would have to contain -CH, as otherwise all Sigma^2_3 sentences would be
Omega-decided by it (since CH implies the quantifier over sets of reals
amounts to a quantifier over subsets of omega_1.) To my mind, however, the
theorem quoted above just says Omega-completeness is not an appropriate
general criterion for theory choice, and so I don't see the logic in then
using it to choose a theory of L(P(omega_1)). (The Pmax axiom does not
give a new domain of conditionally generically absolute statements here,
since it makes the theory of L(P(omega_1)) part of the theory of L(R).)
So in my view, a proof of the Omega-conjecture means we are pretty far
from deciding CH.
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