Theory (mathematical logic)

In mathematical logic, a theory — also called a formal theory — is a special collection of sentences written in a formal language. These sentences follow strict rules that allow us to 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, certain special sentences are chosen as starting points, known as axioms. These axioms are the foundation of the theory. Using the rules of the system, we can prove new sentences called theorems. Every axiom is automatically a theorem because it is already accepted as true without proof.

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

General theories (as expressed in formal language)

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

A theory is a collection of these basic statements. These chosen statements are called the "elementary theorems" of the theory, and they are considered true within that theory. The truth of these statements depends on the theory itself — 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 collection of sentences written in a special kind of language called a first-order formal language. This language helps mathematicians and logicians express ideas clearly and precisely.

There are different ways to show that a sentence follows from the rules of a first-order theory. These methods include Hilbert-style systems, natural deduction, sequent calculus, tableaux method, and resolution. Each method provides a structured way to build proofs and understand the relationships between ideas in the theory.

Examples

One way to create a theory is by choosing a set of axioms in a specific language. These axioms can form the theory by themselves or include their logical consequences. Famous theories made this way are ZFC and Peano arithmetic.

Another way to build a theory is to start with a structure and gather all sentences that match 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.