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  1. Maximality Principles in Set Theory.Luca Incurvati - 2017 - Philosophia Mathematica 25 (2):159-193.
    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 (...)
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  • Set Mapping Reflection.Justin Tatch Moore - 2005 - Journal of Mathematical Logic 5 (1):87-97.
  • Proper Forcing, Cardinal Arithmetic, and Uncountable Linear Orders.Justin Tatch Moore - 2005 - Bulletin of Symbolic Logic 11 (1):51-60.
    In this paper I will communicate some new consequences of the Proper Forcing Axiom. First, the Bounded Proper Forcing Axiom implies that there is a well ordering of R which is Σ 1 -definable in (H(ω 2 ), ∈). Second, the Proper Forcing Axiom implies that the class of uncountable linear orders has a five element basis. The elements are X, ω 1 , ω 1 * , C, C * where X is any suborder of the reals of size (...)
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  • Generic Absoluteness.Joan Bagaria & Sy D. Friedman - 2001 - Annals of Pure and Applied Logic 108 (1-3):3-13.
    We explore the consistency strength of Σ 3 1 and Σ 4 1 absoluteness, for a variety of forcing notions.
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  • Bounded Forcing Axioms and the Continuum.David Asperó & Joan Bagaria - 2001 - Annals of Pure and Applied Logic 109 (3):179-203.
    We show that bounded forcing axioms are consistent with the existence of -gaps and thus do not imply the Open Coloring Axiom. They are also consistent with Jensen's combinatorial principles for L at the level ω2, and therefore with the existence of an ω2-Suslin tree. We also show that the axiom we call BMM3 implies 21=2, as well as a stationary reflection principle which has many of the consequences of Martin's Maximum for objects of size 2. Finally, we give an (...)
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  • On a Convenient Property About $${[\Gamma]^{\Aleph_0}}$$.David Asperó - 2009 - Archive for Mathematical Logic 48 (7):653-677.
    Several situations are presented in which there is an ordinal γ such that ${\{ X \in [\gamma]^{\aleph_0} : X \cap \omega_1 \in S\,{\rm and}\, ot(X) \in T \}}$ is a stationary subset of ${[\gamma]^{\aleph_0}}$ for all stationary ${S, T\subseteq \omega_1}$ . A natural strengthening of the existence of an ordinal γ for which the above conclusion holds lies, in terms of consistency strength, between the existence of the sharp of ${H_{\omega_2}}$ and the existence of sharps for all reals. Also, an (...)
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  • The Bounded Axiom A Forcing Axiom.Thilo Weinert - 2010 - Mathematical Logic Quarterly 56 (6):659-665.
    We introduce the Bounded Axiom A Forcing Axiom . It turns out that it is equiconsistent with the existence of a regular ∑2-correct cardinal and hence also equiconsistent with BPFA. Furthermore we show that, if consistent, it does not imply the Bounded Proper Forcing Axiom.
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  • BPFA and Inner Models.Sy-David Friedman - 2011 - Annals of the Japan Association for Philosophy of Science 19:29-36.
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  • Bounded Namba Forcing Axiom May Fail.Jindrich Zapletal - 2018 - Mathematical Logic Quarterly 64 (3):170-172.
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  • Baumgartnerʼs Conjecture and Bounded Forcing Axioms.David Asperó, Sy-David Friedman, Miguel Angel Mota & Marcin Sabok - 2013 - Annals of Pure and Applied Logic 164 (12):1178-1186.
  • Double Helix in Large Large Cardinals and Iteration of Elementary Embeddings.Kentaro Sato - 2007 - Annals of Pure and Applied Logic 146 (2):199-236.
    We consider iterations of general elementary embeddings and, using this notion, point out helices of consistency-wise implications between large large cardinals.Up to now, large cardinal properties have been considered as properties which cannot be accessed by any weaker properties and it has been known that, with respect to this relation, they form a proper hierarchy. The helices we point out significantly change this situation: the same sequence of large cardinal properties occurs repeatedly, changing only the parameters.As results of our investigation (...)
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  • Subcomplete Forcing Principles and Definable Well-Orders.Gunter Fuchs - 2018 - Mathematical Logic Quarterly 64 (6):487-504.
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  • Strongly Uplifting Cardinals and the Boldface Resurrection Axioms.Joel David Hamkins & Thomas A. Johnstone - 2017 - Archive for Mathematical Logic 56 (7-8):1115-1133.
    We introduce the strongly uplifting cardinals, which are equivalently characterized, we prove, as the superstrongly unfoldable cardinals and also as the almost-hugely unfoldable cardinals, and we show that their existence is equiconsistent over ZFC with natural instances of the boldface resurrection axiom, such as the boldface resurrection axiom for proper forcing.
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  • Forcing Axioms and Ω-Logic.Teruyuki Yorioka - 2009 - Journal of the Japan Association for Philosophy of Science 36 (2):45-52.
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  • Distributive Proper Forcing Axiom and Cardinal Invariants.Huiling Zhu - 2013 - Archive for Mathematical Logic 52 (5-6):497-506.
    In this paper, we study the forcing axiom for the class of proper forcing notions which do not add ω sequence of ordinals. We study the relationship between this forcing axiom and many cardinal invariants. We use typical iterated forcing with large cardinals and analyse certain property being preserved in this process. Lastly, we apply the results to distinguish several forcing axioms.
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  • Resurrection Axioms and Uplifting Cardinals.Joel David Hamkins & Thomas A. Johnstone - 2014 - Archive for Mathematical Logic 53 (3-4):463-485.