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Preservation theorems and restricted consistency statements in bounded arithmetic

Arnold Beckmann Orcid Logo

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

Swansea University Author: Arnold Beckmann Orcid Logo

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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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spelling 2013-10-17T11:50:43.4976257 v2 13722 2012-12-17 Preservation theorems and restricted consistency statements in bounded arithmetic 1439ebd690110a50a797b7ec78cca600 0000-0001-7958-5790 Arnold Beckmann Arnold Beckmann true false 2012-12-17 SCS 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. Journal Article Annals of Pure and Applied Logic 126 1-3 255 280 0168-0072 31 12 2004 2004-12-31 10.1016/j.apal.2003.11.003 COLLEGE NANME Computer Science COLLEGE CODE SCS Swansea University 2013-10-17T11:50:43.4976257 2012-12-17T10:28:52.3612725 Faculty of Science and Engineering School of Mathematics and Computer Science - Computer Science Arnold Beckmann 0000-0001-7958-5790 1
title Preservation theorems and restricted consistency statements in bounded arithmetic
spellingShingle Preservation theorems and restricted consistency statements in bounded arithmetic
Arnold Beckmann
title_short Preservation theorems and restricted consistency statements in bounded arithmetic
title_full Preservation theorems and restricted consistency statements in bounded arithmetic
title_fullStr Preservation theorems and restricted consistency statements in bounded arithmetic
title_full_unstemmed Preservation theorems and restricted consistency statements in bounded arithmetic
title_sort Preservation theorems and restricted consistency statements in bounded arithmetic
author_id_str_mv 1439ebd690110a50a797b7ec78cca600
author_id_fullname_str_mv 1439ebd690110a50a797b7ec78cca600_***_Arnold Beckmann
author Arnold Beckmann
author2 Arnold Beckmann
format Journal article
container_title Annals of Pure and Applied Logic
container_volume 126
container_issue 1-3
container_start_page 255
publishDate 2004
institution Swansea University
issn 0168-0072
doi_str_mv 10.1016/j.apal.2003.11.003
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Computer Science{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Computer Science
document_store_str 0
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description 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.
published_date 2004-12-31T03:17:15Z
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