Axiom

Axiom

An axiom is a statement or proposition that is regarded as being established, accepted, or self-evidently true. In various fields of study, axioms serve as foundational principles from which other statements are logically derived. Axioms are crucial in disciplines such as mathematics, logic, and philosophy.

Mathematics

In mathematics, an axiom is a basic assumption about a mathematical system from which theorems can be logically derived. A well-known example is the axioms of Euclidean geometry, which include statements such as "through any two points, there is exactly one straight line." These axioms form the basis for Euclidean geometry.

Another important set of axioms in mathematics is the Peano axioms, which define the properties of natural numbers. These axioms include the existence of a first natural number (usually 0 or 1) and the principle of mathematical induction.

Logic

In logic, an axiom is a statement that is taken to be true within the system of logic. Logical axioms are used to derive other truths within a logical framework. For example, in propositional logic, the statement "if P implies Q and P is true, then Q is true" is considered an axiom.

Philosophy

In philosophy, axioms are often seen as self-evident truths that do not require proof. Philosophers such as Aristotle and René Descartes have discussed the nature and role of axioms in philosophical inquiry. Descartes' famous statement "I think, therefore I am" is often considered an axiom in his philosophical system.

Related Concepts

• Postulate: A statement assumed to be true without proof, often used interchangeably with axiom.
• Theorem: A statement that has been proven on the basis of previously established statements, such as axioms.
• Lemma: A subsidiary or intermediate theorem in an argument or proof.
• Corollary: A statement that follows readily from a previously proven statement.

Related Pages

• Euclidean geometry
• Peano axioms
• Propositional logic
• Aristotle
• René Descartes