Editing Gödel's completeness theorem (section)

===As a theorem of arithmetic===

The model existence theorem and its proof can be formalized in the framework of [[Peano arithmetic]]. Precisely, we can systematically define a model of any consistent effective first-order theory ''T'' in Peano arithmetic by interpreting each symbol of ''T'' by an arithmetical formula whose free variables are the arguments of the symbol. (In many cases, we will need to assume, as a hypothesis of the construction, that ''T'' is consistent, since Peano arithmetic may not prove that fact.) However, the definition expressed by this formula is not recursive (but is, in general, [[arithmetical hierarchy#The arithmetical hierarchy of sets of natural numbers|Δ<sub>2</sub>]]).