In this Appendix we present the ultrapower construction of a non-standard model of the true arithmetic. The construction is quite elementary and intuitive, and we present it in a self-contained manner (proofs of all statements are included). Indeed, we also discuss a few other basic notions and terminology useful when dealing with non-standard models. All this material is included to help a reader with less experience in mathematical logic to create a mental image of model M that forms the ambient universe for our constructions (Section 1.1).

For the presentation of the construction we fix the language to be LPA, the language of Peano arithmetic (Section 5.1), having constants 0 and 1, two binary functions x + y and x · y, and the binary relation x < y. When talking about arithmetic in mathematical logic this is the usual choice of language. In Section 1.1 we opted for a much larger language Lall as the language of M; we shall discuss the differences after Theorem A.9 but for now we want to keep the whole picture as elementary as possible.

Each symbol from LPA has a canonical interpretation in the set of natural numbers N. We shall denote the resulting LPA-structure also N. The phrase the true arithmetic means (N), the LPA-theory of N, i.e. the set of all LPA-sentences true in N. The structure N is called the standard model of arithmetic.