On embedding models of arithmetic of cardinality $\aleph _1$ into reduced powers

Juliette Kennedy ¹   ; Saharon Shelah ²

¹ Department of Mathematics University of Helsinki P.O. Box 4 FI-00014 University of Helsinki Finland
² Institute of Mathematics Hebrew University 91904 Jerusalem, Israel Department of Mathematics Rutgers University New Brunswick, NJ 08903, U.S.A.

Fundamenta Mathematicae, Tome 176 (2003) no. 1, pp. 17-24

Résumé

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 $.

Détail

DOI : 10.4064/fm176-1-2

Keywords: early tennenbaum proved countable models forall mathbb embeddable reduced product mathbb omega cal where cal cofinite filter paper model forall mathbb aleph embeddable mathbb omega where regular filter omega