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Extendible sets in Peano arithmetic


Author: Stuart T. Smith
Journal: Trans. Amer. Math. Soc. 316 (1989), 337-367
MSC: Primary 03C62; Secondary 03H15
DOI: https://doi.org/10.1090/S0002-9947-1989-0946223-4
MathSciNet review: 946223
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Abstract: Let $ \mathcal{A}$ be a structure and let $ U$ be a subset of $ \vert\mathcal{A}\vert$. We say $ U$ is extendible if whenever $ \mathcal{B}$ is an elementary extension of $ \mathcal{A}$, there is a $ V \subseteq \vert\mathcal{B}\vert$ such that $ (\mathcal{A},U) \prec (\mathcal{B},V)$. Our main results are: If $ \mathcal{M}$ is a countable model of Peano arithmetic and $ U$ is a subset of $ \vert\mathcal{M}\vert$, then $ U$ is extendible iff $ U$ is parametrically definable in $ \mathcal{M}$. Also, the cofinally extendible subsets of $ \vert\mathcal{M}\vert$ are exactly the inductive subsets of $ \vert\mathcal{M}\vert$. The end extendible subsets of $ \vert\mathcal{M}\vert$ are not completely characterized, but we show that if $ \mathcal{N}$ is a model of Peano arithmetic of arbitrary cardinality and $ U$ is any bounded subset of $ \mathcal{N}$, then $ U$ is end extendible.


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DOI: https://doi.org/10.1090/S0002-9947-1989-0946223-4
Article copyright: © Copyright 1989 American Mathematical Society

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