Extendible sets in Peano arithmetic

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.