In 1985 the second author showed that if there is a proper
class of measurable Woodin cardinals and V𝔹1 and V𝔹2 are generic extensions of V satisfying CH then V𝔹1 and V𝔹2 agree on all Σ21-statements. In
terms of the strong logic Ω-logic this can be reformulated by
saying that under the above large cardinal assumption ZFC + CH is
Ω-complete for Σ21. Moreover, CH is the unique
Σ21-statement with this feature in the sense that any other
Σ21-statement with this feature is Ω-equivalent to
CH over ZFC. It is natural to look for other strengthenings
of ZFC that have an even greater degree of Ω-completeness.
For example, one can ask for recursively enumerable axioms A such
that relative to large cardinal axioms ZFC + A is Ω-complete
for all of third-order arithmetic. Going further, for each
specifiable segment Vλ of the universe of sets (for example, one
might take Vλ to be the least level that satisfies there is
a proper class of huge cardinals), one can ask for recursively
enumerable axioms A such that relative to large cardinal axioms
ZFC + A is Ω-complete for the theory of Vλ. If
such theories exist, extend one another, and are unique in the sense
that any other such theory B with the same level of
Ω-completeness as A is actually Ω-equivalent to A
over ZFC, then this would show that there is a unique
Ω-complete picture of the successive fragments of the
universe of sets and it would make for a very strong case for axioms
complementing large cardinal axioms. In this paper we show that
uniqueness must fail. In particular, we show that if there is one
such theory that Ω-implies CH then there is another that
Ω-implies ¬CH.
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