Subsets of models of arithmetic (Q1311429)

The paper contains various results concerning end extensions of countable models of PA. It is shown that a subset \(X\) of a countable model \(M\models\text{PA}\) can be coded in an elementary end extension of \(M\) iff \((M,X)\) satisfies a certain strong collection scheme. Examples are constructed to show that the scheme is weaker than full induction, but is not implied by \(\Sigma_ n\) induction scheme for \((M,X)\), for any \(n\). Other results include a proof that if \(X\subseteq M\) is coded in some elementary end extension of \(M\), then it is also coded in some minimal elementary end extension of \(M\), and a proof that every countable model \(M\) of PA has an elementary end extension in which \(M\) is a semiregular, but not regular, cut.