Nonstandard models of the complete natural-number theory

Let $L_{\mathrm{N}}$ have a constant symbol $0$, a unary function symbol $S$ and a binary relation symbol $<$. The standard natural-number structure is $\mathcal{N}=(\omega ,0,S,<)$, with $0=\mathrm{\varnothing }$, $S(n)=n\cup \{n\}$ and $m<n$ meaning $m\in n$, as in The natural numbers $\mathbb{N}$ (von Neumann). Its complete semantic theory is

$\mathrm{Th}(\mathcal{N})=\{\sigma \in {\mathrm{Sent}}_{L_{\mathrm{N}}}:\mathcal{N}\models \sigma \}\mathrm{.}$

This is a set by Separation on sentence codes using Existence and uniqueness of set satisfaction; sentences are assignment independent by Coincidence for term values and satisfaction. It is called complete because for every sentence $\sigma$ exactly one of $\sigma ,\mathrm{\neg }\sigma$ belongs, by the negation clause and ordinary classical reasoning. It is a theory in the sense of Theories, models and semantic consequence. No internally definable truth predicate in the arithmetic structure is asserted.

The numeral $\bar{n}$ is the closed term given by $\bar{0}=0$ and $\overline{n+1}=S(\bar{n})$. This recursion takes place on the set of term codes and is justified by The recursion theorem. In particular $\bar{1}=S(0)$ and $\bar{2}=S(S(0))$; the numerals denote $0,1,2$ respectively, and term evaluation by Term denotation inductively gives ${\bar{n}}^{\mathcal{N}}=n$ for every $n$.

A nonstandard model of this complete natural-number theory is a nonempty set $L_{\mathrm{N}}$-structure $\mathcal{M}\models \mathrm{Th}(\mathcal{N})$ that is not isomorphic to $\mathcal{N}$. This definition does not assert existence. The language is deliberately $(0,S,<)$ rather than the larger arithmetic language used in the cited source; the complete theory and the meaning of nonstandard are relative to this declared language.