Definition 2.2.12 (Representation of a total function). Let $f:{\mathbb{N}}^k\to \mathbb{N}$ be total and $T$ be any $L_{\mathrm{PA}}$-theory extending PA${}^{-}$. We say that $f$ is represented in $T$ if there is an $L_{\text{PA}{}^{-}}$ formula $𝜃(x_1,\dots ,x_k,y)$ such that, for all $\overline{n}\in {\mathbb{N}}^k$:

• (a)
  $T⊢\exists !y.𝜃(\overline{n},y)$
• (b)
  If $k=f(\overline{n})$, then $T⊢𝜃(\overline{n},\underline{k})$