Nonstandard definability

Stuart T. Smith

doi:10.1016/0168-0072(89)90064-X

Nonstandard definability

Stuart T. Smith

Research output: Contribution to journal › Article › peer-review

21 Scopus citations

Abstract

We investigate the notion of definability with respect to a full satisfaction class σ for a model M of Peano arithmetic. It is shown that the σ-definable subsets of M always include a class which provides a satisfaction definition for standard formulas. Such a class is necessarily proper, therefore there exist recursively saturated models with no full satisfaction classes. Nonstandard extensions of overspill and recursive saturation are utilized in developing a criterion for nonstandard definability. Finally, these techniques yield some information concerning the extendibility of full satisfaction classes from one model to another.

Original language	English
Pages (from-to)	21-43
Number of pages	23
Journal	Annals of Pure and Applied Logic
Volume	42
Issue number	1
DOIs	https://doi.org/10.1016/0168-0072(89)90064-X
State	Published - 28 Mar 1989

ASJC Scopus subject areas

Logic

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10.1016/0168-0072(89)90064-X

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abstract = "We investigate the notion of definability with respect to a full satisfaction class σ for a model M of Peano arithmetic. It is shown that the σ-definable subsets of M always include a class which provides a satisfaction definition for standard formulas. Such a class is necessarily proper, therefore there exist recursively saturated models with no full satisfaction classes. Nonstandard extensions of overspill and recursive saturation are utilized in developing a criterion for nonstandard definability. Finally, these techniques yield some information concerning the extendibility of full satisfaction classes from one model to another.",

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AB - We investigate the notion of definability with respect to a full satisfaction class σ for a model M of Peano arithmetic. It is shown that the σ-definable subsets of M always include a class which provides a satisfaction definition for standard formulas. Such a class is necessarily proper, therefore there exist recursively saturated models with no full satisfaction classes. Nonstandard extensions of overspill and recursive saturation are utilized in developing a criterion for nonstandard definability. Finally, these techniques yield some information concerning the extendibility of full satisfaction classes from one model to another.

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DO - 10.1016/0168-0072(89)90064-X

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Nonstandard definability

Abstract

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