Extendible sets in Peano arithmetic
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- by Stuart T. Smith PDF
- Trans. Amer. Math. Soc. 316 (1989), 337-367 Request permission
Abstract:
Let $\mathcal {A}$ be a structure and let $U$ be a subset of $|\mathcal {A}|$. We say $U$ is extendible if whenever $\mathcal {B}$ is an elementary extension of $\mathcal {A}$, there is a $V \subseteq |\mathcal {B}|$ 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 $|\mathcal {M}|$, then $U$ is extendible iff $U$ is parametrically definable in $\mathcal {M}$. Also, the cofinally extendible subsets of $|\mathcal {M}|$ are exactly the inductive subsets of $|\mathcal {M}|$. The end extendible subsets of $|\mathcal {M}|$ 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.References
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Additional Information
- © Copyright 1989 American Mathematical Society
- 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