On the Consistency of a Slight (?) Modification of Quine’s New Foundations

Jensen, Ronald Björn

doi:10.1007/978-94-010-1709-1_16

Cited by 53 publications

(74 citation statements)

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“…Jensen's pioneering work [11] unveiled a magical link between models of NFU and automorphisms of models of ZF-style set theories, a link that has captured the imagination of other researchers, e.g., Holmes [9], Solovay [24], and two of the authors of the present paper ( [3,16]). Our results here have a number of implications for NFU, for example, Theorem 5.15 can be used to show that every countable model of ZFC can be realized as the strongly cantorian part of a model of NFU.…”

Section: Introductionmentioning

confidence: 69%

Largest initial segments pointwise fixed by automorphisms of models of set theory

2017

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Section: Introductionmentioning

confidence: 69%

Largest initial segments pointwise fixed by automorphisms of models of set theory

2017

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“…Perhaps the most famous of them is NFU, so called "NF with Urelements", introduced by Jensen [7,1969 ], which results from NF by restricting extensionality to non-empty sets. There are also many extensional consistent subsystems of NF.…”

Section: Extensionality Is An Axiommentioning

confidence: 99%

Consistency of strictly impredicative NF and a little more …

Tupailo¹

2010

J. symb. log.

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An instance of Stratified Comprehensionis called strictly impredicative iff, under minimal stratification, the type of x is 0. Using the technology of forcing, we prove that the fragment of NF based on strictly impredicative Stratified Comprehension is consistent. A crucial part in this proof, namely showing genericity of a certain symmetric filter, is due to Robert Solovay.As a bonus, our interpretation also satisfies some instances of Stratified Comprehension which are not strictly impredicative. For example, it verifies existence of Frege natural numbers.Apparently, this is a new subsystem of NF shown to be consistent. The consistency question for the whole theory NF remains open (since 1937).

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“…It will sometimes turn on informal considerations, such as whether one has a good feel for the sorts of things that 33 There is, on the other hand, a consistency proof for NFU (New Foundations with Urelements). See (Jensen, 1969). 34 A set of axioms A is conservative over a theory T relative to mathematical vocabulary V not occurring in T just in case: if ψ is a logical consequence of A ∪ T and V does not occur in ψ, then ψ is a logical consequence of T alone.…”

Section: Further Issuesmentioning

confidence: 99%

On Specifying Truth-Conditions

Rayo¹

2008

The Philosophical Review

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Consider a committalist-someone who believes that assertions of a sentence like 'the number of the planets is 8' carry commitment to numbers-and a noncommittalist-someone who believes that all it takes for assertions of this sentence to be correct is that there be eight planets. The committalist wants to know more about the noncommittalist's view. She understands what the noncommittalist thinks is required of the world in order for assertions of simple sentences like 'the number of the planets is 8' to be correct, but she wants to know how the proposal is supposed to work in general. Could the noncommittalist respond by supplying a recipe for translating each arithmetical sentence into a sentence that wears its ontological innocence on its sleave? Surprisingly, there is a precise and interesting sense in which the answer is 'no'. We will see that when certain constraints are in place, it is impossible to specify an adequate translation-method.Fortunately, there is a technique for specifying truth-conditions that is not based on translation, and can be used to explain to the committalist what the noncommittalist thinks is required of the world in order for arithmetical assertions to be correct. I call it 'the φ(w)-technique'. A shortcoming of the φ(w)-technique is that it is of limited dialectical effectiveness. From the perspective of a committalist, the φ(w)-technique only delivers suitable noncommittalist truth-conditions on the assumption that numbers exist. So the φ(w)-technique is not particularly promising as a maneuver for convincing a committalist to give up mathematical Platonism. But I hope it will earn its keep by providing the committalist with an illuminating account of the noncommittalist position, and by clarifying the role of noncommittalism in addressing certain philosophical puzzles.The φ(w)-technique turns out to have applications beyond arithmetic. It can be used to shed light on the truth-conditions of second-order and set-theoretic sentences. It can also * For their many helpful comments I am indebted to

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On the Consistency of a Slight (?) Modification of Quine’s New Foundations

Cited by 53 publications

References 4 publications

Largest initial segments pointwise fixed by automorphisms of models of set theory

Largest initial segments pointwise fixed by automorphisms of models of set theory

Consistency of strictly impredicative NF and a little more …

On Specifying Truth-Conditions

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