Theory (mathematical logic)

In mathematical logic, a theory — also called a formal theory — is a special group of sentences written in a formal language. These sentences follow strict rules that help us build new truths from old ones. When we add these rules of reasoning to our language, we create something called a formal system.

Inside this system, some special sentences are picked as starting points, called axioms. These axioms are the base of the theory. Using the rules of the system, we can prove new sentences called theorems. Every axiom is also a theorem because it is already accepted as true.

One common type of theory is a first-order theory. This theory uses a set of rules, called inference rules, to build theorems step by step from its axioms. Theories help mathematicians learn what can be proven and what cannot, making them useful tools for studying the basics of mathematics.

General theories (as expressed in formal language)

When we talk about theories in math and logic, we start with some basic ideas or "statements." These statements are like the building blocks of the theory.

A theory is just a group of these basic statements. These special statements are called the "elementary theorems" of the theory, and they are seen as true inside that theory. Whether these statements are true depends on the theory — the same statement might be true in one theory but false in another.

First-order theories

Further information: List of first-order theories

A first-order theory is a set of sentences written in a special language called a first-order formal language. This language helps mathematicians and logicians share ideas clearly.

There are many ways to check if a sentence fits the rules of a first-order theory. These methods include Hilbert-style systems, natural deduction, sequent calculus, tableaux method, and resolution. Each method offers a way to create proofs and see how ideas in the theory connect.

Examples

One way to create a theory is by choosing a set of axioms in a special language. These axioms can form the theory by themselves or include ideas that follow from them. Famous theories made this way are ZFC and Peano arithmetic.

Another way to build a theory is to start with a structure and collect all sentences that fit that structure. For example, using the natural numbers with addition and multiplication gives the theory of true arithmetic, while using all real numbers with these operations creates the theory of real closed fields.