Non-standard model of arithmetic

Example

Let $\mathsf{PA}$ be first-order Peano arithmetic in the language with $0$, $S$, $+$, $\cdot$ and $<$. A non-standard model of arithmetic is a model of $\mathsf{PA}$ that is not isomorphic to the standard model $\N$ of the natural numbers.

Such a model exists. Add a new constant symbol $c$, and let $T = \mathsf{PA} \cup \set{c > \underline{n} : n \in \N}$, where $\underline{n}$ is the term $S S \cdots S 0$ with $n$ copies of $S$. Every finite subset of $T$ mentions only finitely many $n$, so it has a model: $\N$ with $c$ read as a large enough number. By the compactness theorem, $T$ has a model $M$. In $M$ the element $c^M$ is greater than the value of every numeral, so it is not of the form $\underline{n}^M$. In a model isomorphic to $\N$ every element is the value of a numeral, so $M$ is a non-standard model of $\mathsf{PA}$. By the downward Löwenheim–Skolem theorem, $M$ can be taken countable.