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  1. Set-theoretic geology.Gunter Fuchs, Joel David Hamkins & Jonas Reitz - 2015 - Annals of Pure and Applied Logic 166 (4):464-501.
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  • Classes and truths in set theory.Kentaro Fujimoto - 2012 - Annals of Pure and Applied Logic 163 (11):1484-1523.
    This article studies three most basic systems of truth as well as their subsystems over set theory ZF possibly with AC or the axiom of global choice GC, and then correlates them with subsystems of Morse–Kelley class theory MK. The article aims at making an initial step towards the axiomatic study of truth in set theory in connection with class theory. Some new results on the side of class theory, such as conservativity, forcing and some forms of the reflection principle, (...)
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  • The downward directed grounds hypothesis and very large cardinals.Toshimichi Usuba - 2017 - Journal of Mathematical Logic 17 (2):1750009.
    A transitive model M of ZFC is called a ground if the universe V is a set forcing extension of M. We show that the grounds ofV are downward set-directed. Consequently, we establish some fundamental theorems on the forcing method and the set-theoretic geology. For instance, the mantle, the intersection of all grounds, must be a model of ZFC. V has only set many grounds if and only if the mantle is a ground. We also show that if the universe (...)
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  • The hyperuniverse program.Tatiana Arrigoni & Sy-David Friedman - 2013 - Bulletin of Symbolic Logic 19 (1):77-96.
    The Hyperuniverse Program is a new approach to set-theoretic truth which is based on justifiable principles and leads to the resolution of many questions independent from ZFC. The purpose of this paper is to present this program, to illustrate its mathematical content and implications, and to discuss its philosophical assumptions.
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  • What is the theory without power set?Victoria Gitman, Joel David Hamkins & Thomas A. Johnstone - 2016 - Mathematical Logic Quarterly 62 (4-5):391-406.
    We show that the theory, consisting of the usual axioms of but with the power set axiom removed—specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the assertion that every set can be well‐ordered—is weaker than commonly supposed and is inadequate to establish several basic facts often desired in its context. For example, there are models of in which ω1 is singular, in which every set of reals is countable, yet ω1 exists, in which there are sets of (...)
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  • The Ground Axiom.Jonas Reitz - 2007 - Journal of Symbolic Logic 72 (4):1299 - 1317.
    A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of ZFC has a class forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion V=HOD that every set is ordinal definable, and the existence of measurable and supercompact cardinals. The related Bedrock Axiom, asserting that the (...)
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  • Characterizations of pretameness and the Ord-cc.Peter Holy, Regula Krapf & Philipp Schlicht - 2018 - Annals of Pure and Applied Logic 169 (8):775-802.
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  • The Hyperuniverse Project and Maximality.Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo (eds.) - 2018 - Basel, Switzerland: Birkhäuser.
    This collection documents the work of the Hyperuniverse Project which is a new approach to set-theoretic truth based on justifiable principles and which leads to the resolution of many questions independent from ZFC. The contributions give an overview of the program, illustrate its mathematical content and implications, and also discuss its philosophical assumptions. It will thus be of wide appeal among mathematicians and philosophers with an interest in the foundations of set theory. The Hyperuniverse Project was supported by the John (...)
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  • Degree structures: Local and global investigations.Richard A. Shore - 2006 - Bulletin of Symbolic Logic 12 (3):369-389.
    The occasion of a retiring presidential address seems like a time to look back, take stock and perhaps look ahead.Institutionally, it was an honor to serve as President of the Association and I want to thank my teachers and predecessors for guidance and advice and my fellow officers and our publisher for their work and support. To all of the members who answered my calls to chair or serve on this or that committee, I offer my thanks as well. Your (...)
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  • On Sequences of Degrees of Constructibility (Solution of Friedman'S Problem 75).Bohuslav Balcar & Petr Hájek - 1978 - Mathematical Logic Quarterly 24 (19-24):291-296.
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  • Class forcing, the forcing theorem and Boolean completions.Peter Holy, Regula Krapf, Philipp Lücke, Ana Njegomir & Philipp Schlicht - 2016 - Journal of Symbolic Logic 81 (4):1500-1530.
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