Skolem functions

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.