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Remarks on Herbrand normal forms and Herbrand realizations

Remarks on Herbrand normal forms and Herbrand realizations LetA H be the Herbrand normal form ofA andA H,D a Herbrand realization ofA H. We show (i) There is an example of an (open) theory ℐ+ with function parameters such that for someA not containing function parameters[Figure not available: see fulltext.] (ii) Similar for first order theories ℐ+ if the index functions used in definingA H are permitted to occur in instances of non-logical axiom schemata of ℐ, i.e. for suitable ℐ,A[Figure not available: see fulltext.] (iii) In fact, in (1) we can take for ℐ+ the fragment (Σ 1 0 -IA)+ of second order arithmetic with induction restricted toΣ 1 0 -formulas, and in (2) we can take for ℐ the fragment (Σ 1 0,b -IA) of first order arithmetic with induction restricted to formulas VxA(x) whereA contains only bounded quantifiers. (iv) On the other hand, $$PA^2 \vdash A^H \Rightarrow PA \vdash A,$$ wherePA 2 is the extension of first order arithmeticPA obtained by adding quantifiers for functions andA∈ℒ(PA). This generalizes to extensional arithmetic in the language of all finite types but not to sentencesA with positively occurring existential quantifiers for functions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Mathematical Logic Springer Journals

Remarks on Herbrand normal forms and Herbrand realizations

Archive for Mathematical Logic , Volume 31 (5) – Mar 23, 2005

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References (26)

Publisher
Springer Journals
Copyright
Copyright © 1992 by Springer-Verlag
Subject
Mathematics; Mathematical Logic and Foundations; Mathematics, general; Algebra
ISSN
0933-5846
eISSN
1432-0665
DOI
10.1007/BF01627504
Publisher site
See Article on Publisher Site

Abstract

LetA H be the Herbrand normal form ofA andA H,D a Herbrand realization ofA H. We show (i) There is an example of an (open) theory ℐ+ with function parameters such that for someA not containing function parameters[Figure not available: see fulltext.] (ii) Similar for first order theories ℐ+ if the index functions used in definingA H are permitted to occur in instances of non-logical axiom schemata of ℐ, i.e. for suitable ℐ,A[Figure not available: see fulltext.] (iii) In fact, in (1) we can take for ℐ+ the fragment (Σ 1 0 -IA)+ of second order arithmetic with induction restricted toΣ 1 0 -formulas, and in (2) we can take for ℐ the fragment (Σ 1 0,b -IA) of first order arithmetic with induction restricted to formulas VxA(x) whereA contains only bounded quantifiers. (iv) On the other hand, $$PA^2 \vdash A^H \Rightarrow PA \vdash A,$$ wherePA 2 is the extension of first order arithmeticPA obtained by adding quantifiers for functions andA∈ℒ(PA). This generalizes to extensional arithmetic in the language of all finite types but not to sentencesA with positively occurring existential quantifiers for functions.

Journal

Archive for Mathematical LogicSpringer Journals

Published: Mar 23, 2005

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