Beginning of stability theory for Polish Spaces

TL;DR

This paper explores stability theory within Polish spaces and definable structures, establishing key equivalences and demonstrating the existence of indiscernibles, thus providing evidence for a coherent stability framework in these contexts.

Contribution

It introduces equivalent conditions for -stability and proves the existence of indiscernibles, advancing the understanding of stability in Polish and definable structures.

Findings

Established equivalent conditions for -stability.

Proved the existence of indiscernibles under certain conditions.

Provided evidence supporting the development of a stability theory for Polish spaces.

Abstract

We consider stability theory for Polish spaces and more generally for definable structures (say, with elements of a set of reals). We clarify by proving some equivalent conditions for $\aleph_0$-stability. We succeed to prove existence of indiscernibles under reasonable conditions; this gives strong evidence that such a theory exists.