 In detail, regarding the first order Peano axioms (henceforth PA), the existence of non-standard models of them actually follows from completeness together with compactness. One constructs these models, which contain infinitely large integers, as follows: add a new constant symbol c to the language of arithmetic. Extend PA to a new theory PA* by adding to it the infinite collection of axioms: {c &gt; 0, c &gt; 1, …}, where, e.g., 3 is S(S(S(0))). PA* is finitely consistent (i.e., every finite subset of PA* is consistent) hence consistent, hence by the Completeness Theorem it has a model.