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September 2008 A note on standard systems and ultrafilters
Fredrik Engström
J. Symbolic Logic 73(3): 824-830 (September 2008). DOI: 10.2178/jsl/1230396749

Abstract

Let (M,𝒳) ⊨ ACA0 be such that P𝒳, the collection of all unbounded sets in 𝒳, admits a definable complete ultrafilter and let T be a theory extending first order arithmetic coded in 𝒳 such that M thinks T is consistent. We prove that there is an end-extension N ⊨ T of M such that the subsets of M coded in N are precisely those in 𝒳. As a special case we get that any Scott set with a definable ultrafilter coding a consistent theory T extending first order arithmetic is the standard system of a recursively saturated model of T.

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Fredrik Engström. "A note on standard systems and ultrafilters." J. Symbolic Logic 73 (3) 824 - 830, September 2008. https://doi.org/10.2178/jsl/1230396749

Information

Published: September 2008
First available in Project Euclid: 27 December 2008

zbMATH: 1161.03023
MathSciNet: MR2444270
Digital Object Identifier: 10.2178/jsl/1230396749

Rights: Copyright © 2008 Association for Symbolic Logic

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Vol.73 • No. 3 • September 2008
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