[FOM] Question about Woodin Conjecture
Harvey Friedman
friedman at math.ohio-state.edu
Mon May 26 09:41:13 EDT 2003
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.
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.
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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