Finite Embeddability of Sets and Ultrafilters

TL;DR

This paper explores the concept of finite embeddability between sets and ultrafilters on natural numbers, revealing connections with algebraic, topological, and nonstandard models of arithmetic.

Contribution

It introduces and analyzes finite embeddability for sets and ultrafilters, linking it to the structure of the Stone-Cech compactification and nonstandard models.

Findings

Connections between finite embeddability and Stone-Cech algebraic structure

Relations to topological properties of ultrafilters

Links with nonstandard models of arithmetic

Abstract

A set A of natural numbers is finitely embeddable in another such set B if every finite subset of A has a rightward translate that is a subset of B. This notion of finite embeddability arose in combinatorial number theory, but in this paper we study it in its own right. We also study a related notion of finite embeddability of ultrafilters on the natural numbers. Among other results, we obtain connections between finite embeddability and the algebraic and topological structure of the Stone-Cech compactification of the discrete space of natural numbers. We also obtain connections with nonstandard models of arithmetic.