Indefiniteness in Semi-Intuitionistic Set Theories: On a Conjecture of Feferman

Rathjen, Michael

doi:10.1017/jsl.2015.55

Cited by 11 publications

(7 citation statements)

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“…(P3) would then be interpreted as something like "there is a way to transform any proof of WO(R) into a proof that J can be iterated along R." We would thus avoid the need for a determinate totality of sets, as desired. 36 Another option is to develop a realizability interpretation, on which (P3) would say that there is an effective way to transform a realizer of WO(R) into a realizer 35 See Rathjen [37] for an interesting discussion of Feferman's proposal. 36 It might be noted that the (BHK) semantics itself has an impredicative feel to it, since it talks about proofs about proofs (see [43]).…”

Section: ✷∃Xmentioning

confidence: 99%

“…36 Another option is to develop a realizability interpretation, on which (P3) would say that there is an effective way to transform a realizer of WO(R) into a realizer 35 See Rathjen [37] for an interesting discussion of Feferman's proposal. 36 It might be noted that the (BHK) semantics itself has an impredicative feel to it, since it talks about proofs about proofs (see [43]).…”

Section: ✷∃Xmentioning

confidence: 99%

“…2 This approach is not even mentioned in his survey article on predicativity [10], but is discussed in [12] and a number of more recent, though unpublished, manuscripts. See also Rathjen [36,37]. 3 Our analysis and interpretation of Poincaré is inspired by Crosilla [3], as well as by Kreisel [24].…”

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confidence: 99%

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Predicativism as a Form of Potentialism

Linnebo¹,

Shapiro²

2021

The Review of Symbolic Logic

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In the literature, predicativism is connected not only with the Vicious Circle Principle but also with the idea that certain totalities are inherently potential. To explain the connection between these two aspects of predicativism, we explore some approaches to predicativity within the modal framework for potentiality developed in Linnebo (2013) and Linnebo and Shapiro (2019). This puts predicativism into a more general framework and helps to sharpen some of its key theses.

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Section: ✷∃Xmentioning

confidence: 99%

Section: ✷∃Xmentioning

confidence: 99%

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Predicativism as a Form of Potentialism

Linnebo¹,

Shapiro²

2021

The Review of Symbolic Logic

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“…(3) Semi-intuitionistic versions of KP have become important in Feferman's work in connection with discussions of definiteness of concepts and the continuum hypothesis (cf. [17,18,19,20,51]).…”

Section: The Provably Total Set Functions Of Kpmentioning

confidence: 99%

Classifying the provably total set functions of KP and KP(P)

Cook¹,

Rathjen²

2016

Preprint

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This article is concerned with classifying the provably total set-functions of Kripke-Platek set theory, KP, and Power Kripke-Platek set theory, KP(P), as well as proving several (partial) conservativity results. The main technical tool used in this paper is a relativisation technique where ordinal analysis is carried out relative to an arbitrary but fixed set x.A classic result from ordinal analysis is the characterisation of the provably recursive functions of Peano Arithmetic, PA, by means of the fast growing hierarchy [10]. Whilst it is possible to formulate the natural numbers within KP, the theory speaks primarily about sets. For this reason it is desirable to obtain a characterisation of its provably total set functions. We will show that KP proves the totality of a set function precisely when it falls within a hierarchy of set functions based upon a relativised constructible hierarchy stretching up in length to any ordinal below the Bachmann-Howard ordinal. As a consequence of this result we obtain that IKP + ∀x∀y (x ∈ y ∨ x / ∈ y) is conservative over KP for Π 2 -formulae, where IKP stands for intuitionistic Kripke-Platek set theory.In a similar vein, utilising [56], it is shown that KP(P) proves the totality of a set function precisely when it falls within a hierarchy of set functions based upon a relativised von Neumann hierarchy of the same length. The relativisation technique applied to KP(P) with the global axiom of choice, AC global , also yields a parameterised extension of a result in [58], showing that KP(P) + AC global is conservative over KP(P) + AC and CZF + AC for Π P 2 statements. Here AC stands for the ordinary axiom of choice and CZF refers to constructive Zermelo-Fraenkel set theory.

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“…This set theory is related to Solomon Feferman's “semi‐constructive set theory”; see Rathjen (, Sections 1 and 2) for a useful explanation and further references, including to Feferman's (published and unpublished) works. In particular, in both theories, global quantification is intuitionistic, while quantification restricted to any set is classical.…”

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confidence: 99%

Dummett on Indefinite Extensibility

Linnebo

2018

Philosophical Issues

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Dummetts notion of indefinite extensibility is influential but obscure. The notion figures centrally in an alternative Dummettian argument for intuitionistic logic and anti-realism, distinct from his more famous, meaningtheoretic arguments to the same effect. Drawing on ideas from Dummett, a precise analysis of indefinite extensibility is proposed. This analysis is used to reconstruct the poorly understood alternative argument. The plausibility of the resulting argument is assessed.

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Indefiniteness in Semi-Intuitionistic Set Theories: On a Conjecture of Feferman

Abstract: The paper proves a conjecture of Solomon Feferman concerning the indefiniteness of the continuum hypothesis relative to a semi-intuitionistic set theory.

Cited by 11 publications

References 22 publications

Predicativism as a Form of Potentialism

Predicativism as a Form of Potentialism

Classifying the provably total set functions of KP and KP(P)

Dummett on Indefinite Extensibility

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