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Complete Recursive Axiomatizations
Theorem: Let a set of formulas be a recursive 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 other words, if a complete theory has a recursively enumerable axiomatization, then this theory is decidable.
(Note: a finite axiomatization is recursively enumerable. Typical axiomatizations that use “axiom schemas” are also recursively enumerable.)
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.
Proof. Suppose is a complete recursive axiomatization. There are two cases, depending on whether is consistent.
Case 1): The set is inconsistent, that is, there are not models for . Then is the set of all first-order sentences and there is a trivial algorithm for checking whether : always return true.
Case 2): The set is consistent. Given , to check whether we run in parallel two complete theorem provers (which exist by Herbrand theorem), the first one trying to prove the formula , the second one trying to prove the formula ; the procedure terminates if any of these theorem provers succeed (the theorem provers simultaneously searches for longer and longer proofs from a larger and larger finite subsets of ). We show that this is an algorithm that decides .
- Because either or and theorem provers are complete one of these theorem provers will eventually halt. The procedure is therefore an algorithm.
- If is proved, then by soundness of theorem prover, .
- If is proved, then by soundness of theorem prover . Because is consistent, there is a model for . Then is true in , so is false in . Because there is a model where does not hold, we have .
End of Proof.
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