How do you prove the decidability of the empty theory and theory of linear orders?
Uh, what is the "empty theory"?
By empty theory I mean the theory In the empty language ( ie no non logical symbols ) with no axioms.
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.
Separate names with a comma.