Theory (mathematical logic)

A theory is a set of ideas. It uses special rules. These rules help find new facts. We call these new facts theorems.

Some theories start with axioms. Axioms are basic ideas. They are true right away. Every axiom is also a theorem.

A theory can be small. We call a small theory a subtheory. A larger theory is an extension.

Some theories are complete. This means they tell us everything. They can say if a fact is true or false.

Math uses these ideas to study numbers. It helps us find what is true.

A theory is a group of sentences. These sentences use a special language. We use rules to find new facts. These facts are called theorems.

Many theories start with axioms. Axioms are basic ideas. They are true right away. Every axiom is a theorem. A system that uses axioms is called an axiomatic system.

Some theories are consistent. This means they do not prove things that clash. A theory is not consistent if it proves a fact and its opposite. A theory is also called satisfiable if it has a model. A model is a way to show the theory is true.

We can also talk about complete theories. A complete theory tells us everything. It can decide if every sentence is true or false. If it cannot do this, it is an incomplete theory.

Some theories are small. We call a small theory a subtheory. A larger theory is an extension. Math uses these tools to study numbers. It helps us know what is true.

In the world of math, a theory is a collection of sentences. These sentences use a special, formal language. A theory helps us decide which statements are true. You can think of it like a set of rules for a game. Once the rules are set, you can find new facts. These new facts are called theorems.

Many theories work using a step-by-step method. This is often called a deductive system. You start with a group of basic ideas called axioms. These axioms are true right away without any proof. From these axioms, you use rules to find more theorems. If a theory is built this way, it is an axiomatic system. Every single axiom is also considered a theorem.

Mathematicians have studied these systems for a long time. Haskell Curry wrote about the foundations of mathematical logic in 2010. He explained how we build these theories from elementary statements. There are also special ways to build theories using structures. For example, you can build a theory using natural numbers. You can also build one using real numbers.

There are important rules to keep a theory working well. A consistent theory is one that does not clash. It must not prove a sentence and its opposite at once. A theory is also called satisfiable if it has a model. A model is a structure that makes every sentence true. Some theories are complete and can decide every sentence. Incomplete theories cannot decide if every sentence is true or false.

We can also compare different theories to each other. A subtheory is a smaller part of a larger theory. An extension is a larger theory that contains a smaller one. We can use an interpretation to link a theory to a subject. This helps us see how the math relates to the real world. This is like deciding what the word "he" means in a story. Without that link, we cannot know if a sentence is true.

In mathematical logic, a theory is a specific set of sentences within a formal language. It acts as a framework for determining which statements are true within a given system. To build a theory, one must first define a non-empty conceptual class of statements. These initial statements are known as primitive or elementary statements. A theory can be viewed as a way to designate a subset of these statements as being true. This means the truth of an elementary statement often depends on the theory itself. For example, a statement might be true in one theory but false in another.

Many theories function through a process called a deductive system. This system combines a formal language with specific rules for deduction. Within these systems, we often identify a subset of statements called axioms. When a theory is built from these axioms, it is called an axiomatic system. Every axiom is automatically considered a theorem of that theory. A theorem is any element that is deductively closed within the theory. In a first-order theory, theorems are obtained by applying inference rules to the set of axioms.

There are different ways to categorize the relationship between theories. If one theory is a subset of another, it is called a subtheory. The larger theory is then known as an extension or a supertheory. Deductive theories are a special type where the content is based on a formal deductive system. In these theories, any sentence that is a logical consequence of the axioms is also included. This means if a finite set of axioms leads to a certain result, that result is a sentence of the theory.

Mathematicians also use the concept of interpretation to connect theories to subject matter. An interpretation establishes a correspondence between the theory's statements and a specific subject. A full interpretation occurs when every elementary statement has a correspondent in the subject matter. If only some statements have correspondents, it is a partial interpretation. This process provides the semantics, or meaning, for the formulas. A model is a specific type of interpretation where every formula in the theory is satisfied.

To ensure a theory is useful, it must meet certain standards of consistency and completeness. A syntactically consistent theory is one where it is impossible to prove every sentence in the language. In many systems, this means you cannot prove both a sentence and its negation. A satisfiable theory is one that possesses a model to make its sentences true. While all satisfiable theories are consistent, this is not always true in all logics. For instance, in second-order logic, some consistent theories are not satisfiable.

Completeness is another vital property of a theory. A complete consistent theory is one where, for every sentence, either the sentence or its negation is provable. This allows the theory to decide the truth of every statement in its language. If a consistent theory cannot decide every sentence, it is called an incomplete theory. These distinctions help logicians understand the limits and powers of different mathematical systems. Understanding these boundaries is essential for foundational mathematical work.

We can also associate theories directly with mathematical structures. The complete theory of a structure, denoted as Th(A), includes all first-order sentences satisfied by that structure. For example, one can build a theory using the natural numbers or the real numbers. The theory of true arithmetic involves the structure of natural numbers with addition and multiplication. Interestingly, the theory of real numbers was shown by Tarski to be decidable. This means there is a systematic way to determine the truth of its sentences.

Finally, first-order theories are a major area of study with many derivation systems. These systems include the Hilbert-style deductive systems and natural deduction. Other methods include the sequent calculus, the tableaux method, and resolution. A formula is a syntactic consequence of a theory if it can be derived using non-logical axioms. This formal approach allows mathematicians to explore complex ideas like the Compactness theorem or the Löwenheim–Skolem theorem. These tools help define the very boundaries of mathematical thought.