axiomákból

Axioms are fundamental statements or propositions that are considered to be self-evidently true and serve as the starting points for logical deductions. In mathematics and philosophy, axioms are the foundational assumptions upon which entire systems of thought are built. They are not proven within the system itself but are accepted as true to enable the derivation of other truths.

The term "axiom" originates from the Greek word "axioma," meaning "that which is thought worthy or fit" or "that which is believed." Historically, Euclid's "Elements" is a prime example of a system built upon axioms, which he termed "postulates" or "common notions." These postulates, such as "a straight line can be drawn between any two points," are foundational to Euclidean geometry.

In modern logic and mathematics, axioms are carefully chosen to be consistent, independent (ideally, though not always strictly enforced), and sufficient to develop a given theory. For instance, Zermelo-Fraenkel set theory, a standard axiomatic system for mathematics, includes axioms like the axiom of extensionality and the axiom of the empty set. These axioms define the properties and behavior of sets, which are the basic building blocks of most modern mathematics. The validity of an axiom is not determined by its truth in some external reality, but by its utility and consistency within the framework of the system it defines.