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On the question of absolute undecidability

In Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Philosophia Mathematica. Association for Symbolic Logic. pp. 153-188 (2010)

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  1. Epistemic Modality, Mind, and Mathematics.Hasen Khudairi - 2021 - Dissertation, University of St Andrews
    This book concerns the foundations of epistemic modality. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality relates to the computational theory of mind; metaphysical modality; the types of mathematical modality; to the epistemic status of undecidable propositions and abstraction principles in the philosophy of mathematics; to the modal profile of rational propositional intuition; and to the (...)
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  • Epistemic Modality and Absolute Decidability.Hasen Khudairi - manuscript
    This paper aims to contribute to the analysis of the nature of mathematical modality, and to the applications of the latter to unrestricted quantification and absolute decidability. Rather than countenancing the interpretational type of mathematical modality as a primitive, I argue that the interpretational type of mathematical modality is a species of epistemic modality. I argue, then, that the framework of two-dimensional semantics ought to be applied to the mathematical setting. The framework permits of a formally precise account of the (...)
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  • Cognitivism About Epistemic Modality.Hasen Khudairi - manuscript
    This paper aims to vindicate the thesis that cognitive computational properties are abstract objects implemented in physical systems. I avail of the equivalence relations countenanced in Homotopy Type Theory, in order to specify an abstraction principle for epistemic intensions. The homotopic abstraction principle for epistemic intensions provides an epistemic conduit into our knowledge of intensions as abstract objects. I examine, then, how intensional functions in Epistemic Modal Algebra are deployed as core models in the philosophy of mind, Bayesian perceptual psychology, (...)
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  • What is Absolute Undecidability?†.Justin Clarke-Doane - 2013 - Noûs 47 (3):467-481.
    It is often alleged that, unlike typical axioms of mathematics, the Continuum Hypothesis (CH) is indeterminate. This position is normally defended on the ground that the CH is undecidable in a way that typical axioms are not. Call this kind of undecidability “absolute undecidability”. In this paper, I seek to understand what absolute undecidability could be such that one might hope to establish that (a) CH is absolutely undecidable, (b) typical axioms are not absolutely undecidable, and (c) if a mathematical (...)
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  • A Verisimilitudinarian Analysis of the Linda Paradox.Gustavo Cevolani, Vincenzo Crupi & Roberto Festa - 2012 - VII Conference of the Spanish Society for Logic, Methodology and Philosphy of Science.
    The Linda paradox is a key topic in current debates on the rationality of human reasoning and its limitations. We present a novel analysis of this paradox, based on the notion of verisimilitude as studied in the philosophy of science. The comparison with an alternative analysis based on probabilistic confirmation suggests how to overcome some problems of our account by introducing an adequately defined notion of verisimilitudinarian confirmation.
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  • Intrinsic Justifications for Large-Cardinal Axioms.Rupert McCallum - forthcoming - Philosophia Mathematica:nkaa038.
    ABSTRACT The aim of the paper is to answer some arguments raised against mathematical structuralism developed by Michael Resnik. These arguments stress the abstractness of mathematical objects, especially their causal inertness, and conclude that mathematical objects, the structures posited by Resnik included, are inaccessible to human cognition. In the paper I introduce a distinction between abstract and ideal objects and argue that mathematical objects are primarily ideal. I reconstruct some aspects of the instrumental practice of mathematics, such as symbolic manipulations (...)
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  • Gödel’s Incompleteness Theorem and the Anti-Mechanist Argument: Revisited.Yong Cheng - 2020 - Studia Semiotyczne 34 (1):159-182.
    This is a paper for a special issue of Semiotic Studies devoted to Stanislaw Krajewski’s paper. This paper gives some supplementary notes to Krajewski’s on the Anti-Mechanist Arguments based on Gödel’s incompleteness theorem. In Section 3, we give some additional explanations to Section 4–6 in Krajewski’s and classify some misunderstandings of Gödel’s incompleteness theorem related to AntiMechanist Arguments. In Section 4 and 5, we give a more detailed discussion of Gödel’s Disjunctive Thesis, Gödel’s Undemonstrability of Consistency Thesis and the definability (...)
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  • The Residual Access Problem.Sharon Berry - manuscript
    A range of current truth-value realist philosophies of mathematics allow one to reduce the Benacerraf Problem to a problem concerning mathematicians' ability to recognize which conceptions of pure mathematical structures are coherent – in a sense which can be cashed out in terms of logical possibility. In this paper I will clarify what it takes to solve this `residual' access problem and then present a framework for solving it.
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  • Отвъд машината на Тюринг: квантовият компютър.Vasil Penchev - 2014 - Sofia: BAS: ISSK (IPS).
    Quantum computer is considered as a generalization of Turing machine. The bits are substituted by qubits. In turn, a "qubit" is the generalization of "bit" referring to infinite sets or series. It extends the consept of calculation from finite processes and algorithms to infinite ones, impossible as to any Turing machines (such as our computers). However, the concept of quantum computer mets all paradoxes of infinity such as Gödel's incompletness theorems (1931), etc. A philosophical reflection on how quantum computer might (...)
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  • Gödel and Set Theory.Akihiro Kanamori - 2007 - Bulletin of Symbolic Logic 13 (2):153-188.
    Kurt Gödel with his work on the constructible universeLestablished the relative consistency of the Axiom of Choice and the Continuum Hypothesis. More broadly, he ensured the ascendancy of first-order logic as the framework and a matter of method for set theory and secured the cumulative hierarchy view of the universe of sets. Gödel thereby transformed set theory and launched it with structured subject matter and specific methods of proof. In later years Gödel worked on a variety of set theoretic constructions (...)
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  • The Search for New Axioms in the Hyperuniverse Programme.Claudio Ternullo & Sy-David Friedman - 2016 - In Andrea Sereni & Francesca Boccuni (eds.), Objectivity, Realism, and Proof. FilMat Studies in the Philosophy of Mathematics. Berlin: Springer. pp. 165-188.
    The Hyperuniverse Programme, introduced in Arrigoni and Friedman (2013), fosters the search for new set-theoretic axioms. In this paper, we present the procedure envisaged by the programme to find new axioms and the conceptual framework behind it. The procedure comes in several steps. Intrinsically motivated axioms are those statements which are suggested by the standard concept of set, i.e. the `maximal iterative concept', and the programme identi fies higher-order statements motivated by the maximal iterative concept. The satisfaction of these statements (...)
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  • Ipotesi del Continuo.Claudio Ternullo - 2017 - Aphex 16.
    L’Ipotesi del Continuo, formulata da Cantor nel 1878, è una delle congetture più note della teoria degli insiemi. Il Problema del Continuo, che ad essa è collegato, fu collocato da Hilbert, nel 1900, fra i principali problemi insoluti della matematica. A seguito della dimostrazione di indipendenza dell’Ipotesi del Continuo dagli assiomi della teoria degli insiemi, lo status attuale del problema è controverso. In anni più recenti, la ricerca di una soluzione del Problema del Continuo è stata anche una delle ragioni (...)
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  • Is the Continuum Hypothesis a Definite Mathematical Problem?Solomon Feferman - manuscript
    The purpose of this article is to explain why I believe that the Continuum Hypothesis (CH) is not a definite mathematical problem. My reason for that is that the concept of arbitrary set essential to its formulation is vague or underdetermined and there is no way to sharpen it without violating what it is supposed to be about. In addition, there is considerable circumstantial evidence to support the view that CH is not definite.
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  • Richness and Reflection.Neil Barton - 2016 - Philosophia Mathematica 24 (3):330-359.
    A pervasive thought in contemporary philosophy of mathematics is that in order to justify reflection principles, one must hold universism: the view that there is a single universe of pure sets. I challenge this kind of reasoning by contrasting universism with a Zermelian form of multiversism. I argue that if extant justifications of reflection principles using notions of richness are acceptable for the universist, then the Zermelian can use similar justifications. However, I note that for some forms of richness argument, (...)
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  • Against the Judgment-Dependence of Mathematics and Logic.Alexander Paseau - 2012 - Erkenntnis 76 (1):23-40.
    Although the case for the judgment-dependence of many other domains has been pored over, surprisingly little attention has been paid to mathematics and logic. This paper presents two dilemmas for a judgment-dependent account of these areas. First, the extensionality-substantiality dilemma: in each case, either the judgment-dependent account is extensionally inadequate or it cannot meet the substantiality condition (roughly: non-vacuous specification). Second, the extensionality-extremality dilemma: in each case, either the judgment-dependent account is extensionally inadequate or it cannot meet the extremality condition (...)
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  • Computing the Uncomputable; or, The Discrete Charm of Second-Order Simulacra.Matthew W. Parker - 2009 - Synthese 169 (3):447-463.
    We examine a case in which non-computable behavior in a model is revealed by computer simulation. This is possible due to differing notions of computability for sets in a continuous space. The argument originally given for the validity of the simulation involves a simpler simulation of the simulation, still further simulations thereof, and a universality conjecture. There are difficulties with that argument, but there are other, heuristic arguments supporting the qualitative results. It is urged, using this example, that absolute validation, (...)
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  • Modal Structuralism and Reflection.Sam Roberts - 2019 - Review of Symbolic Logic 12 (4):823-860.
    Modal structuralism promises an interpretation of set theory that avoids commitment to abstracta. This article investigates its underlying assumptions. In the first part, I start by highlighting some shortcomings of the standard axiomatisation of modal structuralism, and propose a new axiomatisation I call MSST (for Modal Structural Set Theory). The main theorem is that MSST interprets exactly Zermelo set theory plus the claim that every set is in some inaccessible rank of the cumulative hierarchy. In the second part of the (...)
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  • Strong Logics of First and Second Order.Peter Koellner - 2010 - Bulletin of Symbolic Logic 16 (1):1-36.
    In this paper we investigate strong logics of first and second order that have certain absoluteness properties. We begin with an investigation of first order logic and the strong logics ω-logic and β-logic, isolating two facets of absoluteness, namely, generic invariance and faithfulness. It turns out that absoluteness is relative in the sense that stronger background assumptions secure greater degrees of absoluteness. Our aim is to investigate the hierarchies of strong logics of first and second order that are generically invariant (...)
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  • Incompatible Ω-Complete Theories.Peter Koellner & W. Hugh Woodin - 2009 - Journal of Symbolic Logic 74 (4):1155 - 1170.
    In 1985 the second author showed that if there is a proper class of measurable Woodin cardinals and $V^{B1} $ and $V^{B2} $ are generic extensions of V satisfying CH then $V^{B1} $ and $V^{B2} $ agree on all $\Sigma _1^2 $ -statements. In terms of the strong logic Ω-logic this can be reformulated by saying that under the above large cardinal assumption ZFC + CH is Ω-complete for $\Sigma _1^2 $ Moreover. CH is the unique $\Sigma _1^2 $ -statement (...)
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  • Mathematical Instrumentalism, Gödel’s Theorem, and Inductive Evidence.Alexander Paseau - 2011 - Studies in History and Philosophy of Science Part A 42 (1):140-149.
    Mathematical instrumentalism construes some parts of mathematics, typically the abstract ones, as an instrument for establishing statements in other parts of mathematics, typically the elementary ones. Gödel’s second incompleteness theorem seems to show that one cannot prove the consistency of all of mathematics from within elementary mathematics. It is therefore generally thought to defeat instrumentalisms that insist on a proof of the consistency of abstract mathematics from within the elementary portion. This article argues that though some versions of mathematical instrumentalism (...)
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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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