## Journal of Symbolic Logic

### On Sets of Relations Definable by Addition

James F. Lynch

#### Abstract

For every $k \in \omega$, there is an infinite set $A_k \subseteq \omega$ and a $d(k) \in \omega$ such that for all $Q_0, Q_1 \subseteq A_k$ where $|Q_0| = |Q_1$ or $d(k) < |Q_0|, |Q_1| < \aleph_0$, the structures $\langle \omega, +, Q_0\rangle$ and $\langle \omega, +, Q_1\rangle$ are indistinguishable by first-order sentences of quantifier depth $k$ whose atomic formulas are of the form $u = v, u + v = w$, and $Q(u)$, where $u, v$, and $w$ are variables.

#### Article information

Source
J. Symbolic Logic, Volume 47, Issue 3 (1982), 659-668.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183741093

Mathematical Reviews number (MathSciNet)
MR666823

Zentralblatt MATH identifier
0504.03013

JSTOR