Finitely Supported Sets Containing Infinite Uniformly Supported Subsets

TL;DR

This paper explores the properties of finitely supported sets within a permutation-based set theory framework, focusing on the existence and characteristics of infinite uniformly supported subsets.

Contribution

It investigates the conditions under which finitely supported sets contain or lack infinite uniformly supported subsets, extending the understanding of atomic sets in this theory.

Findings

Classical atomic sets may or may not contain infinite uniformly supported subsets.

Properties of finitely supported sets vary depending on the presence of such subsets.

The study clarifies the structure of finitely supported sets in the context of permutation actions.

Abstract

The theory of finitely supported algebraic structures represents a reformulation of Zermelo-Fraenkel set theory in which every construction is finitely supported according to the action of a group of permutations of some basic elements named atoms. In this paper we study the properties of finitely supported sets that contain infinite uniformly supported subsets, as well as the properties of finitely supported sets that do not contain infinite uniformly supported subsets. For classical atomic sets, we study whether they contain or not infinite uniformly supported subsets.