A simple algebraic characterization of nonstandard extensions (Q2845565)

Heuristically speaking, the nonstandard extension of a set \(X\) is an extension of \(X\) to a superset \(X^*\) together with a way of assigning to every function \(f:X\to X\) an extension \(f^*:X^*\to X^*\) that is, in some sense, ``truth-preserving''. The proper framework for nonstandard extensions is first-order logic but there has been a recent trend to try and give ``logic-free'' presentations of the subject. This article falls into the aforementioned trend by proving that, if the map \(f\mapsto f^*:X^X\to (X^*)^{X^*}\) satisfies three natural axioms, then the map is a nonstandard extension. The way this is achieved is by showing that the axioms imply that the structure \((X^*;(f^*)_{f\in X^X})\) is isomorphic to a \textit{limit ultrapower} of the structure \((X;(f)_{f\in X^X})\); since limit ultrapowers are known to be nonstandard extensions (in fact, they characterize nonstandard extensions by a result of Keisler), this establishes the main result of the paper.NEWLINENEWLINEA word on the aforementioned axioms: Two of the axioms are quite natural and essentially say that the map \(f\mapsto f^*\) respects composition and equality. The third axioms is a little less natural. It states that, given any two \(\xi,\eta\in X^*\), there exist \(f,g\in X^X\) and \(\zeta\in X^*\) such that \(f^*(\zeta)=\xi\) and \(g^*(\zeta)=\eta\). Really, the third axiom states that a certain ordering on nonstandard models, namely the Puritz ordering, is directed. The Puritz order corresponds to the well-known Rudin-Keisler order on ultrafilters.