Theory (mathematical logic)

Background

In mathematical logic, a theory is a set of sentences in a formal language. Typically, a deductive system is assumed, creating a formal system of language and rules.

A sentence that belongs to a deductively closed theory is called a theorem. Many deductive systems have a subset called the axioms, making it an axiomatic system; every axiom is automatically a theorem. A first-order theory consists of first-order sentences derived recursively by applying inference rules to the axioms.

Did You Know?

• The compactness theorem, deduction theorem, and Löwenheim–Skolem theorem are topics related to first-order theories.
• A direct way to specify a theory is to define a set of axioms in a formal language; the theory then includes those axioms and their provable consequences.
• A second way to specify a theory is to begin with a structure and let the theory be the set of sentences that are satisfied by the structure.