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2017 Decidable Fragments of the Simple Theory of Types with Infinity and NF
Anuj Dawar, Thomas Forster, Zachiri McKenzie
Notre Dame J. Formal Logic 58(3): 433-451 (2017). DOI: 10.1215/00294527-2017-0009

Abstract

We identify complete fragments of the simple theory of types with infinity (TSTI) and Quine’s new foundations (NF) set theory. We show that TSTI decides every sentence ϕ in the language of type theory that is in one of the following forms:

(A) ϕ=x1r1xkrky1s1ylslθ where the superscripts denote the types of the variables, s1>>sl, and θ is quantifier-free,

(B) ϕ=x1r1xkrky1sylsθ where the superscripts denote the types of the variables and θ is quantifier-free.

This shows that NF decides every stratified sentence ϕ in the language of set theory that is in one of the following forms:

(A) ϕ=x1xky1ylθ where θ is quantifier-free and ϕ admits a stratification that assigns distinct values to all of the variables y1,,yl,

(B) ϕ=x1xky1ylθ where θ is quantifier-free and ϕ admits a stratification that assigns the same value to all of the variables y1,,yl.

Citation

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Anuj Dawar. Thomas Forster. Zachiri McKenzie. "Decidable Fragments of the Simple Theory of Types with Infinity and NF." Notre Dame J. Formal Logic 58 (3) 433 - 451, 2017. https://doi.org/10.1215/00294527-2017-0009

Information

Received: 20 June 2014; Accepted: 29 May 2015; Published: 2017
First available in Project Euclid: 21 April 2017

zbMATH: 06761617
MathSciNet: MR3681103
Digital Object Identifier: 10.1215/00294527-2017-0009

Subjects:
Primary: 03E70

Rights: Copyright © 2017 University of Notre Dame

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Vol.58 • No. 3 • 2017
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