Editing Diagonal lemma (section)

=== Representation Theorem ===
Let <math>\mathbb{N}</math> be the set of [[Natural number|natural numbers]]. A [[First-order logic|first-order]] [[Theory (mathematical logic)|theory]] <math>T</math> in the language of arithmetic containing <math>\mathsf{Q}</math> ''represents'' the <math>k</math>-ary recursive function <math>f: \mathbb{N}^k\rightarrow\mathbb{N}</math> if there is a [[First-order logic#Formulas|formula]] <math>\varphi_f(x_1, \dots, x_k, y)</math> in the language of <math>T</math> s.t. for all <math>m_1, \dots, m_k \in \mathbb{N} </math>, if <math>f(m_1, \dots, m_k) =  n</math> then <math>T \vdash \forall y (\varphi_f (\overline{m_1}, \dots, \overline{m_k}, y) \leftrightarrow y = \overline{n} )</math>.

The representation theorem is provable, i.e. every recursive function is representable in <math>T</math>.<ref>See Hinman 2005, Chap 4.6 for additional details and a proof of this theorem.</ref>