[FOM] Question about Woodin Conjecture

[Harvey Friedman]

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.

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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