Theory

Theory has a number of distinct meanings in different fields of knowledge, depending on the context and their methodologies.

Mathematics

In mathematics, the word theory is used informally to refer to certain distinct bodies of knowledge about mathematics. This knowledge consists of axioms, definitions, theorems and computational techniques, all related in some way by tradition or practice. Examples include group theory, set theory, Lebesgue integration theory and field theory.

Related Topics:
Mathematics - Group theory - Set theory - Lebesgue integration theory - Field theory

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The term "theory" also has a formal usage in mathematics, particularly in mathematical logic and model theory. A theory in this sense is a set of statements closed under certain rules of inference. A typical theory will present certain axioms and rules, corresponding to a useful or interesting abstraction, and then derive non-obvious theorems from those axioms. The resulting theorems often provide solutions to real-world problems which correspond to the original abstraction. Obvious examples include arithmetic (abstracting the concept of number), geometry (the concept of space), and probability (the concept of randomness).

Related Topics:
Mathematical logic - Model theory - Closed - Axioms - Abstraction - Theorem - Arithmetic - Geometry - Probability

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However, Gödel's incompleteness theorem shows that no consistent theory capable of defining the concept of natural numbers can derive all true statements about those numbers. This sets a fundamental limit to the applicability of any mathematical system.

Related Topics:
Gödel's incompleteness theorem - Consistent - Natural numbers - True

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