In 1985 the second author showed that if there is a proper class of measurable Woodin cardinals and V-B1 and V-B2 are generic extensions of V satisfying CH then V-B1 and V-B2 agree on all Sigma(2)(1)-statements. In terms of the strong logic Omega-logic this can be reformulated by saying that under the above large cardinal assumption ZFC + CH is Omega-complete for Sigma(2)(1). Moreover, CH is the unique Sigma(2)(1)-statement with this feature in the sense that any other Sigma(2)(1)-statement with this feature is Omega-equivalent to CH over ZFC. It is natural to look for other strengthenings of ZFC that have an even greater degree of Omega-completeness. For example. one can ask for recursively enumerable axioms A such that relative to large cardinal axioms ZFC + A is Omega-complete for all of third-order arithmetic. Going further, for each specifiable segment V;. of the universe of sets (for example, one might take V-lambda 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 Omega-complete for the theory of V-lambda. If such theories exist. extend one another. and are unique in the sense that any other such theory B with the same level of Omega-completeness as A is actually Omega-equivalent to A over ZFC, then this would show that there is a unique Omega-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 Omega-implies CH then there is another that Omega-implies inverted left perpendicularCH.
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