Non-standard arithmetic

Goodstein's result from page 1163 is true for all ordinary integers. But since it is independent of the axioms of arithmetic there must be objects that still satisfy the axioms but for which it is false. It turns out however that any such objects must in effect be infinite. For any set of objects that satisfy the axioms of arithmetic must include all finite ordinary integers, since each of these can be reached just by using Δ repeatedly. And the axioms then turn out to imply that any additional objects must be larger than all these integers—and must therefore be infinite. But for any such truly infinite objects operations like + and × cannot be computed by finite procedures, making it difficult to describe such objects in an explicit way. Ever since the work of Thoralf Skolem in 1933 non-standard models of arithmetic have been discussed, particularly in the context of ultrafilters and constructs like infinite trees. (See also page 1172.)