On the set of atoms

Crabbé, Marcel

doi:10.1093/jigpal/8.6.751

Cited by 4 publications

(11 citation statements)

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“…See Corollary 7.2.22 below. This result has been proved previously in [Crabbé, 2000] using the conventional set-theoretical semantics. In the present categorical setting, this result is transparently immediate.…”

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confidence: 76%

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Self-similarity in the Foundations

Gorbow¹

2019

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Let M, N be L-structures. M is a substructure of N if M ⊆ N and for every constant symbol c, relation symbol R and function symbol f of L, we haveNote that since M is an L-structure, the condition on function symbols implies that M, as a subset of N , is closed under f N . We also say that N is an extension of M. The substructure is proper if its domain is a proper subset of the domain of the extension, in which case we write M < N . Note that M ≤ N ⇔ M < N ∨ M = N , whence this defines a partial order.If a subset S of N is closed under the interpretation of all constant and function symbols of N , then we have a substructure N ↾ S of N , called the restriction of N to S, on the domain S defined by c N ↾ S = c N , R N ↾ S = R N ∩ S arity(R) and f N ↾ S = f N ∩ S arity(f )+1 , for all constant, relation and function symbols, c, R, f , respectively.An embedding f : M → N from an L-structure M to an L-structure N is a function f : M → N , such that for each atomic L-formula φ( x) and for each m ∈ M, we have

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confidence: 76%

“…An immediate corollary of these results is that (I)NF is equiconsistent to (I)NFU + |V | = |P(V )|. For the classical case, this has already been proved in [Crabbé, 2000], but the intuitionistic case appears to be new. Moreover, the result becomes quite transparent in the categorical setting.…”

Section: Introduction For Logiciansmentioning

confidence: 72%

Self-similarity in the Foundations

Gorbow¹

2019

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“…An immediate corollary of these results is that (I)NF is equiconsistent to (I)NFU + |V | = |P(V )|. For the classical case, this has already been proved in [4], but the intuitionistic case appears to be new. Moreover, the result becomes quite transparent in the categorical setting.…”

mentioning

confidence: 72%

“…Interpret M Mor (m) as "m is a disjoint union of three classes A, B and f, such that f is a set of pairs coding a function with domain A and co-domain B". 4 For convenience, we extend the functional notation to m in this setting, i.e., m(x) = df f (x), for all x ∈ A, and we also say that m codes this function/morphism from A to B. 3.…”

Section: Axioms 44 ((I)mlu Classmentioning

confidence: 99%

Algebraic New Foundations

Gorbow¹

2019

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“…Rieger-Bernays permutation technology (adapted to consistency and independence results for NF by Dana Scott ([21]), further applied by C. Ward Henson ([8]), and further adapted to NFU by Marcel Crabbé in [3]) can be used to show that any model of NFU has its urelements homogeneous with respect to any stratified formula in which all the variables u i have the same relative type. RiegerBernays techniques can also be used to produce a model of NFU which is not homogeneous for a specific unstratified first-order formula.…”

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 set of atoms

Cited by 4 publications

References 1 publication

Self-similarity in the Foundations

Self-similarity in the Foundations

Algebraic New Foundations

The Usual Model Construction for NFU Preserves Information

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