Abstract We produce, relative to a backslash textsf upper Z upper F upper C $\textsf {ZFC}$ \textsf Z F C model with a supercompact cardinal, a backslash textsf upper Z upper F upper C $\textsf {ZFC}$ \textsf Z F C model of the Proper Forcing Axiom in which the nonstationary ideal on omega 1 $\omega _1$ ω 1 is upper Pi 1 $\Pi _1$ Π 1 -definable in a parameter from upper H Subscript normal first transfinite cardinal 2 $H_{\aleph _2}$ H ℵ 2 .

Abstract Strong Turing Determinacy, or ${\mathrm {sTD}}$ , is the statement that for every set A of reals, if $\forall x\exists y\geq _T x (y\in A)$ , then there is a pointed set $P\subseteq A$ . We prove the following consequences of Turing Determinacy ( ${\mathrm {TD}}$ ) and ${\mathrm {sTD}}$ over ${\mathrm {ZF}}$ —the Zermelo–Fraenkel axiomatic set theory without the Axiom of Choice: (1) ${\mathrm {ZF}}+{\mathrm {TD}}$ implies $\mathrm {wDC}_{\mathbb {R}}$ —a weaker version of $\mathrm {DC}_{\mathbb {R}}$ . (2) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that every set of reals is measurable and has Baire property. (3) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that every uncountable set of reals has a perfect subset. (4) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that for every set of reals A and every $\epsilon>0$ : (a) There is a closed set $F\subseteq A$ such that $\mathrm {Dim_H}(F)\geq \mathrm {Dim_H}(A)-\epsilon $ , where $\mathrm {Dim_H}$ is the Hausdorff dimension. (b) There is a closed set $F\subseteq A$ such that $\mathrm {Dim_P}(F)\geq \mathrm {Dim_P}(A)-\epsilon $ , where $\mathrm {Dim_P}$ is the packing dimension.

Abstract We prove the following two basis theorems for ${\rm{\Sigma }}_2^1$ -sets of reals: (1) Every nonthin ${\rm{\Sigma }}_2^1$ -set has a perfect ${\rm{\Delta }}_2^1$ -subset if and only if it has a nonthin ${\rm{\Delta }}_2^1$ -subset, and this is equivalent to the statement that there is a nonconstructible real. (2) Every uncountable ${\rm{\Sigma }}_2^1$ -set has an uncountable ${\rm{\Delta }}_2^1$ -subset if and only if either every real is constructible or $\omega _1^L$ is countable. We also apply the method that proves (2) to show that if there is a nonconstructible real, then there is a perfect ${\rm{\Pi }}_2^1$ -set with no nonempty ${\rm{\Pi }}_2^1$ -thin subset, strengthening a result of Harrington [4].

Abstract We present a forcing to obtain a localized version of Local Club Condensation, a generalized Condensation principle introduced by Sy Friedman and the first author in [3] and [5]. This forcing will have properties nicer than the forcings to obtain this localized version that could be derived from the forcings presented in either [3] or [5]. We also strongly simplify the related proofs provided in [3] and [5]. Moreover our forcing will be capable of introducing this localized principle at κ while simultaneously performing collapses to make κ become the successor of any given smaller regular cardinal. This will be particularly useful when κ has large cardinal properties in the ground model. We will apply this to measure how much L-likeness is implied by Local Club Condensation and related principles. We show that Local Club Condensation at κ+ is consistent with ¬☐κ whenever κ is regular and uncountable, generalizing and improving a result of the third author in [14], and that if κ ≥ ω2 is regular, CC(κ+) - Chang’s Conjecture at κ+ - is consistent with Local Club Condensation at κ+, both under suitable large cardinal consistency assumptions.

Abstract The Axiom of Strong Condensation, first introduced by Woodin in [14], is an abstract version of the Condensation Lemma ofL. In this paper, we construct a set-sized forcing to obtain Strong Condensation forH(ω2). As an application, we show that “ZFC + Axiom of Strong Condensation + ”is consistent, which answers a question in [14]. As another application, we give a partial answer to a question of Jech by proving that “ZFC + there is a supercompact cardinal + any ideal onω1which is definable overH(ω2) is not precipitous” is consistent under sufficient large cardinal assumptions.