On embedding models of arithmetic of cardinality $\aleph _1$ into reduced powers
Volume 176 / 2003
Fundamenta Mathematicae 176 (2003), 17-24
MSC: 03C62, 03C20, 03C50.
DOI: 10.4064/fm176-1-2
Abstract
In the early 1970's S. Tennenbaum proved that all countable models of ${\rm PA}^- + \forall _1 -{\rm Th}({\mathbb N})$ are embeddable into the reduced product ${\mathbb N}^\omega /{\cal F}$, where ${\cal F}$ is the cofinite filter. In this paper we show that if $M$ is a model of ${\rm PA}^- + \forall _1 -{\rm Th}({\mathbb N})$, and $|M|=\aleph _1$, then $M$ is embeddable into ${\mathbb N}^\omega /D$, where $D$ is any regular filter on $\omega $.
