## FANDOM

78 Pages

A theory $T$ is said to have Skolem functions if for every formula $\phi(x;y)$ there is a term $t(y)$ such that whenever $M \models T$, $b \in M$, and $\phi(M;b)$ is non-empty, then $t(b) \in \phi(M;b)$.

By the Tarski-Vaught criterion, having Skolem functions is enough to ensure that every substructure of a model is an elementary substructure.

Every structure can be expanded to have Skolem functions, by a process called skolemization. For each formula $\phi(x;y)$, one adds a term $t_\phi$ and chooses $t_\phi(b) \in \phi(M;b)$ arbitrarily, for every $b$ for which $\phi(M;b) \ne \emptyset$. After doing this, new formulas may have appeared, so the process must be iterated $\omega$ times. This process is highly non-canonical, and breaks most model-theoretic properties. It is useful as a tool in proving results like the Downwards Löwenheim-Skolem theorem, and the existence of Ehrenfeucht-Mostowski models.

A theory $T$ is said to have definable Skolem functions if for every formula $\phi(x;y)$ there is a definable function $f(y)$ such that whenever $M \models T$, $b \in M$, and $\phi(M;b)$ is non-empty, then $f(b) \in \phi(M;b)$. This is a weaker condition than having Skolem functions.

An equivalent condition to having definable skolem functions is that every definably closed subset of a model is an elementary substructure.

The theory of algebraically closed fields does not have definable skolem functions, but the theories of real closed fields and $p$-adically closed fields do. Any o-minimal expansion of $RCF$ has definable skolem functions (in fact, has definable choice).

If $T$ has definable Skolem functions, $T^{eq}$ need not have definable Skolem functions. In fact, this happens if and only if $T$ has definable choice, a condition stronger than elimination of imaginaries.