Editing Gödel's incompleteness theorems (section)

=== Effective axiomatization ===

A formal system is said to be ''effectively axiomatized'' (also called ''effectively generated'') if its set of theorems is [[recursively enumerable]]. This means that there is a computer program that, in principle, could enumerate all the theorems of the system without listing any statements that are not theorems. Examples of effectively generated theories include Peano arithmetic and [[Zermelo–Fraenkel set theory]] (ZFC).{{sfn|Franzén|2005|p=112}}

The theory known as [[true arithmetic]] consists of all true statements about the standard integers in the language of Peano arithmetic. This theory is consistent and complete, and contains a sufficient amount of arithmetic. However, it does not have a recursively enumerable set of axioms, and thus does not satisfy the hypotheses of the incompleteness theorems.