User:IssaRice/Computability and logic/Theory
In mathematical logic, theory has several related meanings.
- theory as set of sentences
- theory as set of sentences closed under deduction
- theory as something with more structure: something that has a set of axioms, a set of theorems (or a way of finding out what follows from the axioms), a proof system
the intuitive picture is something like: we fix some background logic (e.g. propositional logic, first-order logic), and then we have some machine that can print a bunch of sentences. we want to name the set of outputs of such a machine. if the machine happens to be "talking" about some mathematical structure we care about (e.g., maybe it's printing sentences in arithmetic like "2+2=4"), then each sentence can be interpreted (by us). but want to make sure to separate (1) the stuff the machine can print out (the theory) vs (2) the stuff we believe about the structure (the set of truths) vs (3) the structure itself.