## 502 – The Löwenheim-Skølem theorem

November 8, 2009

In this note I sketch the proof of the Löwenheim-Skølem (or Löwenheim-Skølem-Tarski) theorem for first order theories. This basic result of model theory is really a consequence of a set theoretic combinatorial lemma, as the proof will demonstrate.

Let ${{\mathcal L}}$ be a first order language, understood as a set of constant, function, and relation symbols. Let

$\displaystyle \kappa_{\mathcal L}=|{\mathcal L}|+\aleph_0,$

so ${\kappa_{\mathcal L}}$ is ${|{\mathcal L}|,}$ unless ${{\mathcal L}}$ is finite, in which case we take ${\kappa_{\mathcal L}=\omega.}$ Talking about ${\kappa_{\mathcal L}}$ rather than ${|{\mathcal L}|}$ simplifies the presentation slightly.

The Löwenheim-Skølem theorem is concerned with the possible infinite sizes of models of first order theories. Of course, a theory ${T}$ could only have finite models; the result does not say anything about ${T}$ if that is the case.

Theorem 1 If ${T}$ is a first order theory in a language ${{\mathcal L},}$ and there is at least one infinite model of ${T,}$ then there are models of ${T}$ of size ${\lambda,}$ for all ${\lambda\ge\kappa_{\mathcal L}.}$

We will prove a more precise statement. Before stating it, note that it is possible to have a theory ${T}$ in some uncountable language ${{\mathcal L}}$ such that ${T}$ has models of certain infinite sizes, but not all. Theorem 1 does not say anything about infinite models of ${T}$ of size ${<\kappa_{\mathcal L}.}$ What cardinals in this range are the possible sizes of models of ${T}$ is actually a rather difficult problem, and we will not address it.