On the consistency of an impredicative subsystem of Quine's NF

Crabbé, Marcel

doi:10.2307/2273386

Cited by 20 publications

(24 citation statements)

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“…As usual for Quine's systems, we need stratification ; we also define a restricted notion thereof, which is motivated by the consideration of “loosely predicative” class existence axioms (cf. [3, Definition 1.1, p. 131]).…”

Section: Background On Sans-serifnfmentioning

confidence: 99%

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A fixed point theory over stratified truth

Cantini

2020

Mathematical Logic Qtrly

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We present a theory of stratified truth ST μ with a μ-operator, where terms representing fixed points of stratified monotone operations are available. We prove that ST μ is relatively intepretable into Quine's NF (or subsystems thereof). The motivation is to investigate a strong theory of truth, which is consistent by means of stratification, i.e., by adopting an implicit type theoretic discipline, and yet is compatible with self-reference (to a certain extent). The present version of ST μ is an enhancement of the theory presented in [2].

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Section: Background On Sans-serifnfmentioning

confidence: 99%

“…By a theorem of Crabbé,

sans-serifNFI

is provably consistent, say, in third order arithmetic (at most) [3, p. 134, Theorem 1]. The details of the (different) consistency proofs for

sans-serifNFI

can be found in [3] and in [8, Theorem 3, p. 187].…”

Section: Background On Sans-serifnfmentioning

confidence: 99%

A fixed point theory over stratified truth

Cantini

2020

Mathematical Logic Qtrly

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“…Remark 3 By a theorem of Crabbè 1982, NFI is provably consistent in third order arithmetic. The details of the (different) consistency proofs for NFI can be found in Crabbè 1982 andHolmes 1995. In order to carry out a Kripke-like construction in the NF-systems and to represent the syntax, we shall essentially exploit Quine's homogeneous pairing operation, which does require extensionality and the existence of a copy of the natural numbers.…”

Section: Other Systemsmentioning

confidence: 99%

Unifying the Philosophy of Truth

Achourioti

Galinon

Martínez-Fernández

et al. 2015

Logic, Epistemology, and the Unity of Science

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Logic, Epistemology, and the Unity of Science aims to reconsider the question of the unity of science in light of recent developments in logic. At present, no single logical, semantical or methodological framework dominates the philosophy of science. However, the editors of this series believe that formal techniques like, for example, independence friendly logic, dialogical logics, multimodal logics, game theoretic semantics and linear logics, have the potential to cast new light on basic issues in the discussion of the unity of science.This series provides a venue where philosophers and logicians can apply specific technical insights to fundamental philosophical problems. While the series is open to a wide variety of perspectives, including the study and analysis of argumentation and the critical discussion of the relationship between logic and the philosophy of science, the aim is to provide an integrated picture of the scientific enterprise in all its diversity.More information about this series at

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“…In fact, it is even inconsistent with one of the other known consistent fragments of NF (predicative NF , described in [4]). We do not expand on this here.…”

Section: Inhomogeneity Of Urelementsmentioning

confidence: 99%

The Usual Model Construction for NFU Preserves Information

Holmes¹

2012

Notre Dame J. Formal Logic

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The "usual" model construction for NFU (Quine's New Foundations with urelements, shown to be consistent by Jensen) starts with a model of the usual set theory with an automorphism that moves a rank (this rank is the domain of the model). "Most" elements of the resulting model of NFU are urelements (it appears that information about their extensions is discarded). The surprising result of this paper is that this information is not discarded at all: the membership relation of the original model (restricted to the domain of the model of NFU ) is definable in the language of NFU . A corollary of this is that the urelements of a model of NFU obtained by the "usual" construction are inhomogeneous: this was the question the author was investigating initially. Other aspects of the mutual interpretability of NFU and a fragment of ZFC are discussed in sufficient detail to place this result in context.

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On the consistency of an impredicative subsystem of Quine's NF

Cited by 20 publications

References 5 publications

A fixed point theory over stratified truth

A fixed point theory over stratified truth

Unifying the Philosophy of Truth

The Usual Model Construction for NFU Preserves Information

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