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The Search for New Axioms in the Hyperuniverse Programme

In Andrea Sereni & Francesca Boccuni (eds.), Objectivity, Realism, and Proof. FilMat Studies in the Philosophy of Mathematics. Berlin: Springer. pp. 165-188 (2016)

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  1. Cantorian Set Theory and Limitation of Size.Michael Hallett - 1984 - Oxford, England: Clarendon Press.
    This volume presents the philosophical and heuristic framework Cantor developed and explores its lasting effect on modern mathematics. "Establishes a new plateau for historical comprehension of Cantor's monumental contribution to mathematics." --The American Mathematical Monthly.
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  • A Logical Journey: From Gödel to Philosophy.Hao Wang - 1996 - Bradford.
    Hao Wang was one of the few confidants of the great mathematician and logician Kurt Gödel. _A Logical Journey_ is a continuation of Wang's _Reflections on Gödel_ and also elaborates on discussions contained in _From Mathematics to Philosophy_. A decade in preparation, it contains important and unfamiliar insights into Gödel's views on a wide range of issues, from Platonism and the nature of logic, to minds and machines, the existence of God, and positivism and phenomenology. The impact of Gödel's theorem (...)
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  • Mathematics in Philosophy: Selected Essays.Charles Parsons - 1983 - Cornell Univ Pr.
    This important book by a major American philosopher brings together eleven essays treating problems in logic and the philosophy of mathematics.
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  • Mathematics Without Numbers: Towards a Modal-Structural Interpretation.Geoffrey Hellman - 1989 - Oxford, England: Oxford University Press.
    Develops a structuralist understanding of mathematics, as an alternative to set- or type-theoretic foundations, that respects classical mathematical truth while ...
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  • From Mathematics to Philosophy.Hao Wang - 1974 - London and Boston: London.
    First published in 1974. Despite the tendency of contemporary analytic philosophy to put logic and mathematics at a central position, the author argues it failed to appreciate or account for their rich content. Through discussions of such mathematical concepts as number, the continuum, set, proof and mechanical procedure, the author provides an introduction to the philosophy of mathematics and an internal criticism of the then current academic philosophy. The material presented is also an illustration of a new, more general method (...)
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  • Foundations Without Foundationalism: A Case for Second-Order Logic.Stewart Shapiro - 1991 - Oxford, England: Oxford University Press.
    The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed description of higher-order logic, including a comprehensive discussion of its semantics. He goes on to demonstrate the prevalence of second-order concepts in mathematics and the extent to which mathematical ideas can be formulated in higher-order logic. He also shows how first-order languages are often insufficient to codify (...)
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  • Set Theory and its Philosophy: A Critical Introduction.Michael D. Potter - 2004 - Oxford, England: Oxford University Press.
    Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set (...)
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  • Naturalism in Mathematics.Penelope Maddy - 1997 - Oxford, England: Oxford University Press.
    Naturalism in Mathematics investigates how the most fundamental assumptions of mathematics can be justified. One prevalent philosophical approach to the problem--realism--is examined and rejected in favor of another approach--naturalism. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be successfully applied in set theory. Her clear, original treatment of this fundamental issue is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both disciplines.
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  • What is Cantor's Continuum Problem?Kurt Gödel - 1947 - In Solomon Feferman, John Dawson & Stephen Kleene (eds.), Journal of Symbolic Logic. Oxford University Press. pp. 176--187.
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  • Multiverse Conceptions in Set Theory.Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo - 2015 - Synthese 192 (8):2463-2488.
    We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’. Furthermore, we propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. In Sect. 2, we use this classification to review four major conceptions. Finally, in Sect. 3, we focus on the distinction between actualism and potentialism with regard to the (...)
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  • On the Question of Absolute Undecidability.Peter Koellner - 2010 - In Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Philosophia Mathematica. Association for Symbolic Logic. pp. 153-188.
    The paper begins with an examination of Gödel's views on absolute undecidability and related topics in set theory. These views are sharpened and assessed in light of recent developments. It is argued that a convincing case can be made for axioms that settle many of the questions undecided by the standard axioms and that in a precise sense the program for large cardinals is a complete success “below” CH. It is also argued that there are reasonable scenarios for settling CH (...)
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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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  • On the Consistency Strength of the Inner Model Hypothesis.Sy-David Friedman, Philip Welch & W. Hugh Woodin - 2008 - Journal of Symbolic Logic 73 (2):391 - 400.
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  • Foundational Implications of the Inner Model Hypothesis.Tatiana Arrigoni & Sy-David Friedman - 2012 - Annals of Pure and Applied Logic 163 (10):1360-1366.
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  • Cantorian Set Theory and Limitation of Size. Michael Hallett.Robert Bunn - 1988 - Philosophy of Science 55 (3):461-478.
    The usual objections to infinite numbers, and classes, and series, and the notion that the infinite as such is self-contradictory, may... be dismissed as groundless. There remains, however, a very grave difficulty, connected with the contradiction [of the class of all classes not members of themselves]. This difficulty does not concern the infinite as such, but only certain very large infinite classes.
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  • Internal Consistency and the Inner Model Hypothesis.Sy-David Friedman - 2006 - Bulletin of Symbolic Logic 12 (4):591-600.
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  • The Role of the Absolute Infinite in Cantor's Conception of Set.Ignacio Jané - 1995 - Erkenntnis 42 (3):375 - 402.
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  • Mathematics in Philosophy: Selected Essays.Charles Parsons - 1985 - British Journal for the Philosophy of Science 36 (4):437-457.
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  • Über Grenzzahlen Und Mengenbereiche: Neue Untersuchungen Über Die Grundlagen der Mengenlehre.Ernst Zermelo - 1930 - Fundamenta Mathematicæ 16:29--47.
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  • Cantorian Set Theory and Limitation of Size.Gregory H. Moore - 1987 - Journal of Symbolic Logic 52 (2):568-570.
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  • Higher-Order Logic Reconsidered.Ignasi Jané - 2005 - In Stewart Shapiro (ed.), Oxford Handbook of Philosophy of Mathematics and Logic. Oxford University Press. pp. 781--810.
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  • Believing the Axioms. I.Penelope Maddy - 1988 - Journal of Symbolic Logic 53 (2):481-511.
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  • The Potential Hierarchy of Sets.Øystein Linnebo - 2013 - Review of Symbolic Logic 6 (2):205-228.
    Some reasons to regard the cumulative hierarchy of sets as potential rather than actual are discussed. Motivated by this, a modal set theory is developed which encapsulates this potentialist conception. The resulting theory is equi-interpretable with Zermelo Fraenkel set theory but sheds new light on the set-theoretic paradoxes and the foundations of set theory.
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  • On Reflection Principles.Peter Koellner - 2009 - Annals of Pure and Applied Logic 157 (2-3):206-219.
    Gödel initiated the program of finding and justifying axioms that effect a significant reduction in incompleteness and he drew a fundamental distinction between intrinsic and extrinsic justifications. Reflection principles are the most promising candidates for new axioms that are intrinsically justified. Taking as our starting point Tait’s work on general reflection principles, we prove a series of limitative results concerning this approach. These results collectively show that general reflection principles are either weak ) or inconsistent. The philosophical significance of these (...)
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  • Cantorian Set Theory and Limitation of Size.Michael Hallett - 1986 - Mind 95 (380):523-528.
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  • Admissible Sets and Structures.Jon Barwise - 1978 - Studia Logica 37 (3):297-299.
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  • From Mathematics to Philosophy.Hao Wang - 1975 - British Journal for the Philosophy of Science 26 (2):170-174.
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  • Sur le platonisme dans les mathématiques.Paul Bernays - 1935 - L’Enseignement Mathematique 34:52--69.
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  • Set Theoretic Naturalism.Penelope Maddy - 1996 - Journal of Symbolic Logic 61 (2):490-514.
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  • Believing the Axioms. II.Penelope Maddy - 1988 - Journal of Symbolic Logic 53 (3):736-764.
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  • Cantorian Set Theory and Limitation of Size.Michael Hallett - 1990 - Studia Logica 49 (2):283-284.
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  • Philosophy of Mathematics.Paul Benacerraf & Hilary Putnam - 1985 - Philosophy of Science 52 (3):488-489.
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  • Mathematics in Philosophy.Charles Parsons - 1987 - Revue Philosophique de la France Et de l'Etranger 177 (1):88-90.
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