How Woodin changed his mind: new thoughts on the Continuum Hypothesis

Rittberg, Colin Jakob

doi:10.1007/s00407-014-0142-8

Cited by 12 publications

(8 citation statements)

References 15 publications

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“…the interpretation of A in M[G]; H(κ) is comprised of all of the sets whose transitive closure is < κ (cf. Rittberg, 2015); and L(R) Pmax |= H(ω 2 ), ∈, I N S , A G |= 'φ'. P is a homogeneous partial order in L(R), such that the generic extension of L(R) P inherits the generic invariance, i.e., the absoluteness, of L(R).…”

Section: Large Cardinalsmentioning

confidence: 99%

Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism

Khudairi

2019

Philosophical Studies Series

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This essay examines the philosophical significance of Ω-logic in Zermelo-Fraenkel set theory with choice (ZFC). The dual isomorphism between algebra and coalgebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The modal profile of Ω-logical validity can then be countenanced within a coalgebraic logic, and Ω-logical validity can be defined via deterministic automata. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal profiles of Ω-logical validity correspond to those of second-order logical consequence, Ω-logical validity is genuinely logical, and thus vindicates a neo-logicist conception of mathematical truth in the set-theoretic multiverse. Second, the foregoing provides a modal-computational account of the interpretation of mathematical vocabulary, adducing in favor of a realist conception of the cumulative hierarchy of sets. * Forthcoming in d'Alfonso and Berkich (eds),'On the Cognitive, Ethical, and Scientific Dimensions of Artificial Intelligence -Themes from IACAP 2016' (Springer: 2018). further be defined via automata. In Section 3, I examine how models of epistemic modal algebras to which modal coalgebraic automata are dually isomorphic are availed of in the computational theory of mind. Finally, in Section 4, the philosophical significance of the characterization of the modal profile of Ωlogical validity for the philosophy of mathematics is examined. I argue (i) that it vindicates a type of neo-logicism with regard to mathematical truth in the set-theoretic multiverse, and (ii) that it provides a modal and computational account of formal grasp of the concept of 'set', adducing in favor of a realist conception of the cumulative hierarchy of sets. Section 5 provides concluding remarks.

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Section: Large Cardinalsmentioning

confidence: 99%

Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism

Khudairi

2019

Philosophical Studies Series

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“…(Woodin 2011, p. 17) As an important corollary, whichever such universe of sets the new axiom picks out would provide a definite answer on claims such as the Continuum Hypothesis. For further details on Woodin's project, see the excellent description in (Rittberg 2015).…”

Section: Looking Forward: Conceptual Engineering For Mathematical Conmentioning

confidence: 99%

Conceptual engineering for mathematical concepts

Tanswell

2017

Inquiry

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In this paper I investigate how conceptual engineering applies to mathematical concepts in particular. I begin with a discussion of Waismann's notion of open texture, and compare it to Shapiro's modern usage of the term. Next I set out the position taken by Lakatos which sees mathematical concepts as dynamic and open to improvement and development, arguing that Waismann's open texture applies to mathematical concepts too. With the perspective of mathematics as open-textured, I make the case that this allows us to deploy the tools of conceptual engineering in mathematics. I will examine Cappelen's recent argument that there are no conceptual safe spaces and consider whether mathematics constitutes a counterexample. I argue that it does not, drawing on Haslanger's distinction between manifest and operative concepts, and applying this in a novel way to set-theoretic foundations. I then set out some of the questions that need to be engaged with to establish mathematics as involving a kind of conceptual engineering. I finish with a case study of how the tools of conceptual engineering will give us a way to progress in the debate between advocates of the Universe view and the Multiverse view in set theory.

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“…Noteworthy contributions to this debate by set theorists include (Baldwin 2018;Antos et al 2015;Shelah 2014;Steel 2014;Väänänen 2014;Magidor 2012;Shelah 2003) and the works of Woodin and Hamkins I will introduce in the following two sections. This debate has been the focus of much of the contemporary philosophizing about set theory, including Barton (2019), Centrone et al 2019, Tanswell (2018), Rittberg (2015), Kennedy (2014), Kennedy and Kossak (2011), Maddy (2011), Koellner (2006).…”

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confidence: 99%

“…In the first decade of this millennium, he gave up this argument and now proposes an argument for an Ultimate-L. In the historically minded (Rittberg 2015), I present both arguments in detail. Here I focus exclusively on the Ultimate-L argument.…”

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confidence: 99%

“…The essence of the argument against the generic-multiverse position is that assuming the Ω Conjecture is true (and that there is a proper class of Woodin cardinals) then this position is simply a brand of formalism that denies the transfinite by a reducing truth about the universe of sets to truth about a simple fragment such as the integers or, in this case, the collection of all subsets of the least Woodin cardinal. (Woodin 2011d, p. 17) For details on the Ω conjecture and the point about Woodin cardinals, see Rittberg (2015).…”

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confidence: 99%

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Mathematical Practices Can Be Metaphysically Laden

Rittberg¹

2020

Handbook of the History and Philosophy of Mathematical Practice

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In this chapter I explore the reciprocal relationship between the metaphysical views mathematicians hold and their mathematical activity. I focus on the settheoretic pluralism debate, in which set theorists disagree about the implications of their formal mathematical work. As a first case study, I discuss how Woodin's monist argument for an Ultimate-L feeds on and is fed by mathematical results and metaphysical beliefs. In a second case study, I present Hamkins' pluralist proposal and the mathematical research projects it endows with relevance. These case studies support three claims: (1) the metaphysical views of mathematicians can shape what counts as relevant research; (2) mathematical results can shape the metaphysical beliefs of mathematicians; (3) metaphysical thought and mathematical activity develop in tandem in mathematical practices. This makes metaphysical thought an integral part of mathematical practices.

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How Woodin changed his mind: new thoughts on the Continuum Hypothesis

Cited by 12 publications

References 15 publications

Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism

Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism

Conceptual engineering for mathematical concepts

Mathematical Practices Can Be Metaphysically Laden

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