## Journal of Symbolic Logic

### Model-Theoretic Properties Characterizing Peano Arithmetic

Richard Kaye

#### Abstract

Let $\mathscr{L} = \{0, 1, +, \cdot, <\}$ be the usual first-order language of arithmetic. We show that Peano arithmetic is the least first-order $\mathscr{L}$-theory containing $I\Delta_0 + \exp$ such that every complete extension $T$ of it has a countable model $K$ satisfying. (i) $K$ has no proper elementary substructures, and (ii) whenever $L \prec K$ is a countable elementary extension there is $\bar{L} \prec L$ and $\bar{K} \subseteq_\mathrm{e} \bar{L}$ such that $K \prec_{\mathrm{cf}}\bar{K}$. Other model-theoretic conditions similar to (i) and (ii) are also discussed and shown to characterize Peano arithmetic.

#### Article information

Source
J. Symbolic Logic, Volume 56, Issue 3 (1991), 949-963.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183743742

Digital Object Identifier
doi:10.2178/jsl/1183743742

Mathematical Reviews number (MathSciNet)
MR1129158

Zentralblatt MATH identifier
0746.03032

JSTOR
links.jstor.org

#### Citation

Kaye, Richard. Model-Theoretic Properties Characterizing Peano Arithmetic. J. Symbolic Logic 56 (1991), no. 3, 949--963. doi:10.2178/jsl/1183743742. https://projecteuclid.org/euclid.jsl/1183743742