The empty theory

[zeberdee]

How do you prove the decidability of the empty theory and theory of linear orders?

 

[HallsofIvy]

Uh, what is the "empty theory"?

 

[zeberdee]

By empty theory I mean the theory In the empty language ( ie no non logical symbols ) with no axioms.

 

[Preno]

The sentences of the empty theory over the empty language include only the logical connective, quantifiers and equality, so deciding them is quite simple. The empty theory over a language which includes only single-argument predicates is monadic logic, which is also decidable. The empty theory over a language which includes at least one predicate with two arguments is, however, not decidable.

The theory of linear orders should be decidable using quantifier elimination.