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.Comment: to appear in Bulletin of the Polish Academy of Sciences, Math Serie
