[FOM] Question about Woodin Conjecture
John Steel
steel at math.berkeley.edu
Mon May 26 14:43:31 EDT 2003
The precise formulation you have given is not correct. See below.
On Mon, 26 May 2003, Harvey Friedman wrote:
> In preparation for an FOM review of Dehornory's "Recent Progress on
> the Continuum Hypothesis (after Woodin)", I would like to get
> absolutely clear about at least some reasonable formulation of
> Conjecture 1. I hope that the experts on the list can inform me
> whether or not I have given a reasonable formulation that is open and
> fully relevant.
>
> From Dehornoy:
>
> Conjecture 1 (Woodin, 1999). Every set theory that is compatible with
> the existence of large cardinals and makes the properties of sets
> with hereditary cardinality at most Aleph1 invariant under forcing
> implies that the Continuum Hypothesis be false.
>
> ****************
>
> Let T be a set of sentences in the language of set theory formulated
> in the ordinary first order predicate calculus with equality, that
> includes the axioms of ZFC. Furthermore assume the following.
>
> 1. T + "there exists arbitrarily large Woodin cardinals" is consistent.
>
This is not enough--we want T to be consistent with
supercompacts, huge cardinals, and all our yet undreamt-of large
cardinal hypotheses of the future. A key move is to take preservation
under set forcing as characteristic of all large cardinal hypotheses,
of today and of tomorrow. The consistency requirement on T then
becomes: true in some set generic extension of V.
Your consistency requirement would be satisfied by T =
"I am the canonical minimal inner model with arbitrarily large
Woodin cardinals".
> 2. Let A be a sentence in the language of set theory. Let A* be the
> sentence resulting from relativizing all quantifiers in A to "sets
> whose transitive closure has cardinality at most Aleph1". Then the
> sentence
>
> if A* then A* holds in all generic extensions of V via a set forcing notion
>
> is provable in T in the sense of ordinary first order predicate
> calculus with equality.
>
You only want to look at generic extensions satisfying T.
(Otherwise, the sentence A* expressing CH will witness that there
is no T of the sort you demand.)
Moreover, provability should be replaced by truth in all such
extensions.
So a precise version of conjecture 1 would
be:
Let T be an axiomatizable theory extending ZFC. Suppose T is true in
some set generic extension of V, and suppose further that if G,H are set
generic over V, and T is true in V[G] and V[H], then (H_omega_2,\in)^V[G]
is elementarily equivalent to (H_omega_2,\in)^V[H].
Then in any set generic extension of V where T is true, CH is false.
(Warning: I forget whether we need to restrict the complexity of
the axioms of T going beyond ZFC, as is necessary in the Omega
conjecture. So in the above, to be safe, restrict the new axioms
of T to be Sigma^m_n, for some m,n.)
This conjecture is meant to be considered under the hypothesis
that there are arbitrarily large Woodin cardinals, though if it were
proved under stronger large cardinal hypotheses the same point would be
made.
The gist of the conjecture is that there is no generically absolute
complete theory of (H_omega_2, \in) which is consistent with all large
cardinal hypotheses and has CH in it. The idea for proving this is to go
through the Omega-conjecture, which implies that there is no generically
absolute complete theory of (V_\omega+2,\in) consistent with all large
cardinals, period.
John Steel
> Then T proves the negation of the continuum hypothesis in the sense
> of ordinary first order predicate calculus with equality.
>
> Harvey Friedman
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