A model of some theory that is not isomorphic to the intended model of the theory. For example, if Peano Arithmetic $\text{PA}$ is consistent, i.e., has a model, then there must exist models other than the set of natural numbers $\mathbb{N}$ with the arithmetical operations. In such models, there exist infinitely many non-standard numbers that are greater than any $n\in \mathbb{N}$. Similarly, there exist models of the real line $ℝ$ that include infinitesimals, i.e., elements greater than $0$ but smaller than any real number $r\in ℝ$. Such non-standard elements can have interesting properties; for example, if $\text{PA}$ is consistent, then $\text{PA}+\neg \text{Con}\left(\text{PA}\right)$ has a model, where $\neg \text{Con}\left(\text{PA}\right)$ the sentence

• • there exists a number $n$ such that $n$ encodes a proof of $0=1$, i.e., $\exists x\text{Proves}\left(x,⌜0=1⌝\right)$

holds, where $\exists x\text{Proves}\left(x,y\right)$ is a provability predicate. But for any natural number $n$, the formula $\text{Proves}\left(n,\text{⌜}0=1\text{⌝}\right)$ is false in this model. Hence, the proof of the inconsistency of $\text{PA}$ must be encoded by a non-standard number, i.e., it must have a non-standard proof.