The **Löwenheim-Skolem Theorem** says that if *M* is an infinite model in some language *L*, then for every cardinal , there is a model *N* of cardinality , 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 containing *S*, with . In particular, taking *S* to be an arbitrary subset of size with , we can find an elementary substructure of *M* of size .

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

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 for every infinite .

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*|.

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

Note that over a set of size , there are at most definable sets. Consequently,

Now given *S ⊆ M* as in the theorem, let

By basic cardinal arithmetic, . Then 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

,

so *e(D)* is an element of . 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

Given an infinite structure *M* and a cardinal 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

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