The [Arithmetic fixed point theorem][1] (see also [MO/30874][2]) states that if $F$ is a formula in number theory with only one free variable $v$, then there is a sentence $A$ such that number theory can prove $A \Leftrightarrow F_v(\underline{[A]})$. An immediate application is [Gödel's Theorem][3].


  [1]: https://en.wikipedia.org/wiki/Diagonal_lemma
  [2]: https://mathoverflow.net/questions/30874
  [3]: https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems