Compactness Theorem

Let be a first-order Theory.
If every finite subset of has a Model, then has a Model.

Proof

If then by Adequacy Theorem
As proofs are finite, there’s a finite s.t. .
Then by the Soundness Theorem i.e. has no model.

Corollary

Finite Groups cannot be axiomatized as a first-order Theory.

Proof

Suppose is a Theory in some Language
whose Models are exactly the finite Groups.
Then let

where

Any finite subset of has a model, e.g. the cyclic for sufficiently large
By compactness, has a Model.
But then this model is infinite.

Corollary

If a first order theory has arbitrarily large finite models,
then it has an infinite model.

Proof

Essentially the same.

Corollary

The Upwards Löwenheim-Skolem Theorem