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 $Q = P \cup \left\{{\neg x = 0, \neg x = s0, \neg x = ss0, \ldots}\right\}$ 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.

$\blacksquare$

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