# Complete Recursive Axiomatizations

Theorem: Let a set of first-order sentences be a recursively enumerable axiomatization for a complete and consistent theory, that is:

• is recursively enumerable: there exists an enumerateion of the set and there exists an algorithm that given computes ;
• is complete: for each FOL sentence , either or ;

Then there exists an algorithm for checking, given , whether .

Proof.

In short: if a complete theory has a recursively enumerable axiomatization, then this theory is decidable.

Conversely: if a theory is undecidable (there is no algorithm for deciding whether a sentence is true or false), then the theory does not have a recursive axiomatization.

Note:

• a finite axiomatization is recursively enumerable
• typical axiomatizations that use “axiom schemas” are also recursively enumerable
• if is an axiomatization for for some interpretation , that is , then is an axiomatization of a complete theory

Corrollary: Let be an interpretation. Then exactly one of the following is true:

• there exists an algorithm for checking
• there is no enumerable set of axioms such that .

Example: the theory of integers with multiplication and quantifiers is undecidable

• consequently, there are no complete axiomatizations for it, no decidable set of axioms from which the truth value of facts about natural numbers follows
• this result is one part of Goedel's incompleteness theorem