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END-EXTENSIONS OF MODELS OF WEAK ARITHMETIC FROM COMPLEXITY-THEORETIC CONTAINMENTS

Published online by Cambridge University Press:  19 July 2016

LESZEK ALEKSANDER KOŁODZIEJCZYK
LESZEK ALEKSANDER KOŁODZIEJCZYK*
Affiliation:
INSTITUTE OF MATHEMATICS UNIVERSITY OF WARSAW BANACHA 2, 02-097 WARSZAWA POLANDE-mail: lak@mimuw.edu.pl
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Abstract

We prove that if the linear-time and polynomial-time hierarchies coincide, then every model of Π1(ℕ) + ¬Ω1 has a proper end-extension to a model of Π1(ℕ), and so Π1(ℕ) + ¬Ω ⊢ BΣ1. Under an even stronger complexity-theoretic assumption which nevertheless seems hard to disprove using present-day methods, Π1(ℕ) + ¬Exp ⊢ BΣ1. Both assumptions can be modified to versions which make it possible to replace Π1(ℕ) by IΔ0 as the base theory.

We also show that any proof that IΔ0 + ¬Exp does not prove a given finite fragment of BΣ1 has to be “nonrelativizing”, in the sense that it will not work in the presence of an arbitrary oracle.

Keywords

collection schemeend-extensionsweak arithmetictime hierarchieslinear time hierarchyfractional-exponential growth
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 3 , September 2016 , pp. 901 - 916
Copyright
Copyright © The Association for Symbolic Logic 2016 

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