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Preservation theorems and restricted consistency statements in bounded arithmetic / Arnold Beckmann

Annals of Pure and Applied Logic, Volume: 126, Issue: 1-3, Pages: 255 - 280

Swansea University Author: Arnold, Beckmann

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Abstract

In this article we prove preservation theorems for theories of bounded arithmetic. The following one is well-known: The ∀Π b 1 - separation of bounded arithmetic theories S i 2 from T j 2 (1 ≤ i ≤ j) is equivalent to the existence of a model of S i 2 which does not have a Δ b 0 - elementary extensio...

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Published in: Annals of Pure and Applied Logic
ISSN: 0168-0072
Published: 2004
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URI: https://cronfa.swan.ac.uk/Record/cronfa13722
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Abstract: In this article we prove preservation theorems for theories of bounded arithmetic. The following one is well-known: The ∀Π b 1 - separation of bounded arithmetic theories S i 2 from T j 2 (1 ≤ i ≤ j) is equivalent to the existence of a model of S i 2 which does not have a Δ b 0 - elementary extension to a model of T j 2 .Let Ω1nst denote that there is a nonstandard element c such that the function n→2log(n)c is a total function.Let BLΣ b 1 be the bounded collection schema ∀x≤|t| ∃y φ(x,y) → ∃z ∀x≤|t| ∃y≤z φ(x,y) for φ ∈ Σ b 1 .Main Theorem. The ∀Π b 1 - separation of S i 2 from T j 2 (1 ≤ i ≤ j) is equivalent to the existence of a model of S i 2 + Ω1nst which is 1b - closed w.r.t. T j 2 , a countable model of S i 2 + BLΣ b 1 without weak end extensions to models of T j 2 .These results still hold when the theories are extended by finitely many ∃ ∀ (Σ b i ∪ Π b i ) - sentences.
College: College of Science
Issue: 1-3
Start Page: 255
End Page: 280