Baire reflection

Abstract

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.

On a fragment of the universal Baire property for $\Sigma^1_2$ sets

“…In the extension V [G] by P we will have Σ 1 2 sets ccc-Universally Baire. Thus Σ 1 2 sets are Lebesgue measurable and have the property of Baire which implies that ω 1 is inaccessible to reals (see [4]). Thus in ω 1 = ω V 2 is weakly compact in L in V [G] by a result of Harrington and Shelah (see [2] or Lemma 7 of [3]).…”

“…2.6 of [6] and Thm. 3.1 of [4]). The place to look for a definable counterexample is the pointclass Σ 1 2 with κ either ω 1 or ω 2 .…”

“…The implication of (1) to (2) is immediate as a club of σ ∈ [κ] ω is closed under a Banach-Mazur strategy in the space κ ω . Regarding the reverse implication, any set of reals of size ω 1 is a counterexample at κ = ω 2 assuming Martin's Maximum (see Theorem 2.6 of [9] and Theorem 3.1 of [7]). The place to look for a definable counterexample is the pointclass Σ 1 2 with κ either ω 1 or ω 2 .…”

“…This equivalence, however, is not true level-by-level. In particular, the relevant part of their argument (originating in [4]) would require that Σ 1 2 sets be ω ω+1 -Universally Baire to prove the existence of 0 # . Using the full strength of Covering for L this can be reduced to ω 3 .…”