Universally Baire Sets of Reals

“…It is straightforward to see (using the absoluteness of well-foundedness) that A G is independent of the choice of T and S. See [4] for further details. …”

mentioning

confidence: 99%

Incompatible Ω-Complete Theories

Koellner¹,

Woodin²

2009

J. symb. log.

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In 1985 the second author showed that if there is a proper class of measurable Woodin cardinals and V B 1 and V B 2 are generic extensions of V satisfying CH then V B 1 and V B 2 agree on all Σ 2 1 -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 Σ 2 1 . Moreover, CH is the unique Σ 2 1 -statement with this feature in the sense that any other Σ 2 1 -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 V λ of the universe of sets (for example, one might take V λ 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 V λ . 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. * We would like to thank the referee for helpful comments.

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“…Fix i < 2. Note that II has complete control over e γ at limit stages by clause (4). II wins by extending arbitrarily at successor stages, and at limits α setting p α (e α (δ α , ξ)) = i for every ξ < ω 1 and, via some simple bookkeeping, choosing the sequence…”

Section: Theorem 12mentioning

confidence: 99%

Baire reflection

Todorčević

Zoble

2008

Trans. Amer. Math. Soc.

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Abstract. We study reflection principles involving nonmeager sets and the Baire Property which are consequences of the generic supercompactness of ω 2 , such as the principle asserting that any point countable Baire space has a stationary set of closed subspaces of weight ω 1 which are also Baire spaces. These principles entail the analogous principles of stationary reflection but are incompatible with forcing axioms. Assuming MM, there is a Baire metric space in which a club of closed subspaces of weight ω 1 are meager in themselves. Unlike stronger forms of Game Reflection, these reflection principles do not decide CH, though they do give ω 2 as an upper bound for the size of the continuum.

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Universally Baire Sets of Reals

Cited by 107 publications

References 14 publications

The Weak Ultrafilter Axiom

The Weak Ultrafilter Axiom

Incompatible Ω-Complete Theories

Baire reflection

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