Existence of Non-Standard Models of Arithmetic

Theorem
There exist non-standard models of arithmetic.

Proof
Let $$P\ $$ be the set of axioms of Peano arithmetic.

Let $$Q = P \cup \{\neg x = 0, \neg x = s0, \neg x = ss0, ... \}$$ where $$x\ $$ is a variable of the language.

Then each finite subset of $$Q$$ is satisfied by the standard model of arithmetic

Hence $$Q$$ is satisfiable by the Compactness theorem.

But any model satisfying $$Q$$ must assign $$x$$ to an element which cannot be obtained by iterating the successor operator on zero a finite number of times.