Teoreemit

Teoreemit are formal statements in mathematics that have been proven to be true based on axioms and previously established results. They serve as the foundational building blocks of mathematical theory, and a teoreem typically asserts a precise relationship or property, with its truth established by a logical argument known as a proof. By contrast, conjectures are unproven statements, while lemmas are auxiliary results used in proving larger theorems, and corollaries are results that follow directly from a teoreem.

Etymology: The term teoreem comes from the Greek theorema, via Latin and other languages; in Estonian mathematical usage teoreem is the standard word for theorem, and teoreemit is the plural.

Historical overview: The concept traces to ancient Greece, with Euclid's Elements presenting many foundational teoreemit. The formal notion of proof and axiomatic system matured in the 19th and 20th centuries, culminating in rigorous axiomatizations of mathematics and logic, such as Hilbert's program.

Examples: Classical teoreemit include the Pythagorean teoreem, which relates the sides of a right triangle; Fermati lõplik teoreem (Fermat's Last Theorem), stating that no three positive integers a, b, c satisfy a^n + b^n = c^n for n > 2; and the Fundamental teoreem of calculus, linking differentiation and integration. In modern logic, Gödel's incompleteness theorems are famous results about the limits of formal systems.

Notable status: Teoreemit are verifiable within an accepted axiomatic framework; their proofs may be simple or highly nontrivial, and a single teoreem can have wide-reaching consequences across multiple fields.