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The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal

Bulletin of Symbolic Logic 8 (1):91-93 (2002)

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Maximality Principles in Set Theory.Luca Incurvati - 2017 - Philosophia Mathematica 25 (2):159-193.details In set theory, a maximality principle is a principle that asserts some maximality property of the universe of sets or some part thereof. Set theorists have formulated a variety of maximality principles in order to settle statements left undecided by current standard set theory. In addition, philosophers of mathematics have explored maximality principles whilst attempting to prove categoricity theorems for set theory or providing criteria for selecting foundational theories. This article reviews recent work concerned with the formulation, investigation and justification (...) of maximality principles. (shrink) Download Export citation Bookmark 11 citations
Ns Saturated and -Definable.Stefan Hoffelner - 2021 - Journal of Symbolic Logic 86 (1):25-59.details We show that under the assumption of the existence of the canonical inner model with one Woodin cardinal$M_1$, there is a model of$\mathsf {ZFC}$in which$\mbox {NS}_{\omega _{1}}$is$\aleph _2$-saturated and${\Delta }_{1}$-definable with$\omega _1$as a parameter which answers a question of S. D. Friedman and L. Wu. We also show that starting from an arbitrary universe with a Woodin cardinal, there is a model with$\mbox {NS}_{\omega _{1}}$saturated and${\Delta }_{1}$-definable with a ladder system$\vec {C}$and a full Suslin treeTas parameters. Both results rely on (...) Download Export citation Bookmark 3 citations
Kurepa trees and Namba forcing.Bernhard König & Yasuo Yoshinobu - 2012 - Journal of Symbolic Logic 77 (4):1281-1290.details We show that strongly compact cardinals and MM are sensitive to $\lambda$-closed forcings for arbitrarily large $\lambda$. This is done by adding ‘regressive' $\lambda$-Kurepa trees in either case. We argue that the destruction of regressive Kurepa trees requires a non-standard application of MM. As a corollary, we find a consistent example of an $\omega_2$-closed poset that is not forcing equivalent to any $\omega_2$-directed-closed poset. Download Export citation Bookmark 4 citations
The restriction of a Borel equivalence relation to a sparse set.Howard Becker - 2003 - Archive for Mathematical Logic 42 (4):335-347.details We consider sparseness, smoothness and the Glimm-Effros Dichotomy for the restriction of a Borel equivalence relation on a Polish space to definable subsets of that space. Download Export citation Bookmark

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