What Can a Categoricity Theorem Tell Us?

Meadows, Toby

doi:10.1017/s1755020313000178

Cited by 11 publications

(5 citation statements)

References 21 publications

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“…This is because for Isaacson's Kreisel, informal rigour is dependent upon the degree to which we have understood a mathematical subject matter. If we expand our concept of set C 0 to one C 1 producing a consistent axiomatisation (as, let's assume, both Ultimate-L and PFA do) our understanding should be cashed out in terms of this new concept C 1 , and this determines (given that we are employing C 1 ) a subject matter that supports either PFA or Ultimate-L, depending on which route we 50 See Meadows (2013) for a survey. Hamkins is also explicit about the point when discussing a version of the categoricity argument in Martin (2001):…”

Section: Objections and Repliesmentioning

confidence: 99%

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Structural Relativity and Informal Rigour

Barton

2012

Boston Studies in the Philosophy and History of Science

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Informal rigour is the process by which we come to understand particular mathematical structures and then manifest this rigour through axiomatisations. Structural relativity is the idea that the kinds of structures we isolate are dependent upon the logic we employ. We bring together these ideas by considering the level of informal rigour exhibited by our set-theoretic discourse, and argue that different foundational programmes should countenance different underlying logics (intermediate between first-and second-order) for formulating set theory. By bringing considerations of perturbations in modal space to bear on the debate, we will suggest that a promising option for representing current set-theoretic thought is given by

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Section: Objections and Repliesmentioning

confidence: 99%

“…54 We might think that this fact has philosophical import. Meadows (in Meadows (2013)) identifies three roles for a categoricity theorem:…”

Section: Objections and Repliesmentioning

confidence: 99%

Structural Relativity and Informal Rigour

Barton

2012

Boston Studies in the Philosophy and History of Science

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“…5 As we shall see, the distinction between P Z-universes and CZ-universes will be important for understanding the sense in which reflection principles can be viewed as axioms for the Zermelian Multiversist. 6 See, for example, [Hamkins, 2012] and [Meadows, 2013]. 7 To avoid ambiguity, I will use the female pronoun for the Zermelian (when Zermelo himself is not being denoted) and the male pronoun to denote the Universist.…”

Section: Introductionmentioning

confidence: 99%

Richness and Reflection

Barton

2015

Philosophia Mathematica

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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, the status of reflection principles as axioms is left open for the Zermelian. * The author is very grateful to 1 A sentence φ is bivalent iff it is either true or false (and not both). There is a rich literature on bivalence in the Philosophy of Mathematics and Philosophy more widely: an excellent recent discussion of some of these issues is available in [Rumfitt, 2015].

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“…Good examples here are so called large cardinal axioms, as well as forcing axioms, and inner model hypotheses.10 Here we are playing slightly fast-and-loose with debates in the foundations of set theory; under a natural interpretation of Joel Hamkins' multiverse perspective, set theory also should be understood as purely algebraic. See[Hamkins, 2012] for the original presentation of this view and[Barton, 2016] for an argument to the effect that this results in a purely algebraic interpretation.11 The exact dialectic import of a categoricity proof is something of a vexed question, see[Meadows, 2013] for discussion. An argument that the quasi-categoricity of ZFC 2 shows that our axiomatisation has been successful is available in[Isaacson, 2011].12 The original proof of this is available in[Zermelo, 1930], and is subsequently tidied up in[Shepherdson, 1951] and[Shepherdson, 1952].…”

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confidence: 99%

Set Theory and Structures

Barton¹,

Friedman²

2019

Synthese Library

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Set-theoretic and category-theoretic foundations represent different perspectives on mathematical subject matter. In particular, category-theoretic language focusses on properties that can be determined up to isomorphism within a category, whereas set theory admits of properties determined by the internal structure of the membership relation. Various objections have been raised against this aspect of set theory in the category-theoretic literature. In this article, we advocate a methodological pluralism concerning the two foundational languages, and provide a theory that fruitfully interrelates a 'structural' perspective to a set-theoretic one. We present a set-theoretic system that is able to talk about structures more naturally, and argue that it provides an important perspective on plausibly structural properties such as cardinality. We conclude the language of set theory can provide useful information about the notion of mathematical structure.

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What Can a Categoricity Theorem Tell Us?

Cited by 11 publications

References 21 publications

Structural Relativity and Informal Rigour

Structural Relativity and Informal Rigour

Richness and Reflection

Set Theory and Structures

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