Boolos on the justification of set theory

Paseau, Alexander

doi:10.1093/philmat/nkl017

Cited by 25 publications

(6 citation statements)

References 13 publications

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“…59 See, for example, (Gödel, 1964, p. 258-259) footnote 16 and Tait (1998). See Paseau (2007) for discussion. 60 See (Gödel, 1964, p. 259) and .…”

Section: Modal Structural Reflection Principlesmentioning

confidence: 99%

Modal Structuralism and Reflection

Roberts

2018

The Review of Symbolic Logic

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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 article, I look at the prospects for supplementing MSST with a modal structural reflection principle, as suggested in Hellman (2015). I show that Hellman’s principle is inconsistent (Theorem 5.32), and argue that modal structural reflection principles in general are either incompatible with modal structuralism or extremely weak.

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“…59 See, for example, (Gödel, 1964, p. 258-259) footnote 16 and Tait (1998). See Paseau (2007) for discussion. 60 See (Gödel, 1964, p. 259) and .…”

Section: Modal Structural Reflection Principlesmentioning

confidence: 99%

Modal Structuralism and Reflection

Roberts

2018

The Review of Symbolic Logic

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“…10 See Diaconescu (1975). See also Boolos (1989) and Paseau (2007) for discussion. For an extensive discussion of Diaconescu's theorem, as well as of the relation between AC and other choice principles on the one hand and classical logic and constructive logics on the other, see DeVidi (2004).…”

Section: Nonclassical Semantics and Standard Set Theorymentioning

confidence: 99%

“…11 The locus classicus is Boolos (1971). See also Boolos (1989) and Paseau (2007) for discussion. 12 Formally: ∀x(x = ∅ → ∃y(y ∈ x ∧ ¬∃z(z ∈ x ∧ z ∈ y))).…”

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confidence: 99%

On Adopting Kripke Semantics in Set Theory

Incurvati

2008

Rev. symb. logic

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Several philosophers have argued that the logic of set theory should be intuitionistic on the grounds that the open-endedness of the set concept demands the adoption of a nonclassical semantics. This paper examines to what extent adopting such a semantics has revisionary consequences for the logic of our set-theoretic reasoning. It is shown that in the context of the axioms of standard set theory, an intuitionistic semantics sanctions a classical logic. A Kripke semantics in the context of a weaker axiomatization is then considered. It is argued that this semantics vindicates an intuitionistic logic only insofar as certain constraints are put on its interpretation. Wider morals are drawn about the restrictions that this places on the shape of arguments for an intuitionistic revision of the logic of set theory. 1 The picture is captured by the following definition by transfinite recursion. Let α be an ordinal.Then, the levels of the hierarchy, V , are organized as follows:

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“…9 I do this without taking a stand on whether there is any good sense in which concepts are a kind of object.10 The latter might be something like Burgess' notion of "heuristic intuition"(Burgess, 2014, p. 27).11 Which is not to say that I think it's correct. See Section 9.12 See alsoBoolos (1971) andPaseau (2007) for more detail. Parsons also lists Tait and Koellner as examples of philosophers who have at least taken the idea that axioms can be conceptually justified in this way seriously(Parsons, 2014, p. 149).…”

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confidence: 99%

“…12 12 See also Boolos (1971) and Paseau (2007) for more detail. Parsons also lists Tait and Koellner as examples of philosophers who have at least taken the idea that axioms can be conceptually justified in this way seriously (Parsons, 2014, p. 149).…”

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confidence: 99%

Gödelian platonism and mathematical intuition

Wrigley

2021

European J of Philosophy

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This paper has two key aims. The first is to clarify the nature of Gödel's platonism. I offer an interpretation of Gödel's remarks on realism and intuition from which a clarified Gödelian position, called conceptual platonism, can be extracted. The second aim is to assess the philosophical merits of this theory. I defend it from charges of mysticism and theology, arguing instead that conceptual platonism is problematic for a more familiar reason. Given the familiarity of this problem, there are implications for a broad family of views in the philosophy of mathematics.

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Boolos on the justification of set theory

Cited by 25 publications

References 13 publications

Modal Structuralism and Reflection

Modal Structuralism and Reflection

On Adopting Kripke Semantics in Set Theory

Gödelian platonism and mathematical intuition

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