Postulaatit

Postulaatit, Finnish for postulates or axioms, are the basic statements that a mathematical or logical theory takes as given without proof. They define the fundamental properties and relations of the objects studied and provide the starting point for deductive reasoning within the system. In a formal theory, postulaatit are distinguished from definitions, which introduce new terms, and from theorems, which are conclusions proven from the postulaatit and previously established results.

Postulaatit can be simple or highly formal. They are often expressed as statements about existence, equality, or incidence and are typically organized into an axiom system or axiom schema. The choice of postulaatit shapes what can be proven and how the objects of the theory behave. A famous example is Euclid's postulates for plane geometry; among them the parallel postulate has inspired the development of non-Euclidean geometries when modified.

In arithmetic and logic, other well-known sets of postulaatit appear, such as the Peano axioms for the natural numbers and the Zermelo–Fraenkel axioms (with or without the axiom of choice) for set theory. These systems illustrate how different collections of postulaatit yield robust foundations for entire branches of mathematics.

The study of postulaatit involves questions of consistency and independence: whether a postulate cannot be proved from the others, and how adding or removing postulaatit affects the theory. The broader project of axiomatization seeks to formalize mathematics around fixed sets of postulaatit, an approach associated with Hilbert and later developments. Different axiom systems can prove the same theorems or give rise to new mathematical frameworks.