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 .
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.
- 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