Löwenheim-Skolem Theorem

The Löwenheim-Skolem Theorem says that if M is an infinite model in some language L, then for every cardinal $$\kappa \ge |L|$$, there is a model N of cardinality $$\kappa$$, elementarily equivalent to M.

More precisely, one has two theorems:

Downward Löwenheim-Skolem Theorem: Let M be an infinite model in some language L. Then for any subset S ⊆ M, there exists an elementary substructure $$N \preceq M$$ containing S, with $$|N| = |S| + |L|$$. In particular, taking S to be an arbitrary subset of size $$\kappa$$ with $$|L| \le \kappa \le |M|$$, we can find an elementary substructure of M of size $$\kappa$$.

Upward Löwenheim-Skolem Theorem: Let M be an infinite model in some language L. Then for every cardinal $$\kappa$$ bigger than |M| and |L|, there is an elementary extension of M of size $$\kappa$$.

On the level of theories, the Löwenheim-Skolem Theorem implies that if T is a theory with an infinite model, then T has a model of cardinality $$\kappa$$ for every infinite $$\kappa \ge |T|$$.

These statements become slightly simpler when working in a countable language. In this case, Upward Löwenheim-Skolem says that if M is an infinite structure, then M has elementary extensions of all cardinalities greater than |M|. Similarly, Downward Löwenheim-Skolem implies that if M is an infinite structure, then M has elementary substructures of all infinite sizes less than |M|.

Proof of Downward Löwenheim-Skolem Theorem
Let M be a structure. For each non-empty definable subset D of M, choose some element e(D) ∈ D, using the axiom of choice. If X is any subset of M, let
 * $$ c(X) = X \cup \{e(D) : D \text{ definable over }X,~ D \ne \emptyset \}$$

Note that over a set of size $$\lambda$$, there are at most $$\lambda + |L|$$ definable sets. Consequently,
 * $$ |c(X)| \le |X| + |L|$$

Now given S ⊆ M as in the theorem, let
 * $$ N = S \cup c(S) \cup c(c(S)) \cup \cdots $$

By basic cardinal arithmetic, $$ |N| = |S| + |L| $$. Then $$ N \preceq M $$ by the Tarski-Vaught test. Indeed, if D is a subset of M definable over N, then D uses only finitely many parameters, and is therefore definable over c(i)(S) ⊆ N for some i. Then
 * $$ e(D) \in c^{(i+1)}(S) \subset N$$,

so e(D) is an element of $$N \cap D$$. Therefore, every non-empty N-definable set intersects N. Therefore the Tarski-Vaught criterion holds and N is an elementary substructure of M. It has the correct size. QED

Proof of Upward Löwenheim-Skolem Theorem
Given an infinite structure M and a cardinal $$\kappa$$ at least as big as both |M| and |L|, let T be the union of the elementary diagram of M and the collection of statements
 * $$ \{c_\alpha \ne c_\beta : \alpha < \beta < \kappa \} $$

where $$\{c_\alpha\}_{\alpha < \kappa}$$ is a collection of $$\kappa$$ new constant symbols. By compactness, T is consistent. Indeed, any finite subset of T only mentions finitely many of the $$c_\alpha$$ and therefore has a model consisting of M with the finitely many $$c_\alpha$$ interpreted as distinct elements of M. So by compactness we can find a model $$N \models T$$. Then N is a model of the elementary diagram of M, so N is an elementary extension of M. Also, the $$c_\alpha$$ ensure that N contains at least $$\kappa$$ distinct elements, i.e., $$|N| \ge \kappa$$. There is a possiblity that N is too big; to hit $$\kappa$$ on the nose, we use Downward Löwenheim-Skolem to find an elementary substructure of N having size $$\kappa$$ and containing M. On general grounds, the resulting structure is an elementary extension of M. QED