About

For questions on axioms, mathematical statements that are accepted as being true, usually without controversy.

An axiom is a premise or starting point of reasoning. As classically conceived, an axiom is a premise so evident as to be accepted as true without controversy.

Axioms define and delimit the realm of analysis. In other words, an axiom is a formal statement that is assumed to be true. Therefore, its truth is taken for granted within the particular domain of analysis, and serves as a starting point for deducing and inferring other (theory and domain dependent) truths. An axiom is defined as a mathematical statement that is accepted as being true without a mathematical proof.

It should be mentioned that in modern times some statements receive a status of axioms, but they are still provable from weaker theories using other statements. One famous example is the axiom of choice, which is provable from ZF set theory if we assume Zorn's lemma. Generally, in modern foundations of mathematics, an axiom is just a statement in the base theory.

history | excerpt history