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