Finite Embeddability of Sets and Ultrafilters

Andreas Blass¹ ; Mauro Di Nasso²

¹ Mathematics Department University of Michigan Ann Arbor, MI 48109, U.S.A.
² Dipartimento di Matematica Università di Pisa 56127 Pisa, Italy

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 63 (2015) no. 3, pp. 195-206.

Résumé

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–Čech compactification of the discrete space of natural numbers. We also obtain connections with nonstandard models of arithmetic.

Détail

DOI : 10.4064/ba8024-1-2016

Keywords: set natural numbers finitely embeddable another set every finite subset has rightward translate subset notion finite embeddability arose combinatorial number theory paper study its own right study related notion finite embeddability ultrafilters natural numbers among other results obtain connections between finite embeddability algebraic topological structure stone ech compactification discrete space natural numbers obtain connections nonstandard models arithmetic