bebizonyítható

Bebizonyítható is a Hungarian adjective meaning "provable" or "demonstrable." In mathematics and formal logic, a statement is bebizonyítható if there exists a finite derivation from the axioms of a formal system that yields the statement as a theorem. The term is used to distinguish what is provable within a given system from what is merely true or convincing in a broader sense.

Morphology: The root bizonyít means "to prove." The prefix be- adds the sense of "capable of" or "established." The suffix -ható marks potential ability, yielding bebizonyítható: "able to be proven."

Usage: In proofs and theorems, bebizonyítható is applied to statements that have formal proofs. For example, in Euclidean geometry, the Pythagorean theorem can be shown bebizonyítható from the axioms of Euclidean geometry. In proof theory, a statement is bebizonyítható if a deduction exists within the chosen formal system; if no such deduction exists, the statement is not bebizonyítható in that system.

Relation to truth and decidability: Provability is system-dependent. Gödel's incompleteness theorems show that in any sufficiently powerful consistent system, there are true statements about the natural numbers that are not bebizonyítható within the system, highlighting a gap between bebizonyítható and absolute truth.

See also: provability, proof theory, formal systems, Gödel's incompleteness theorems.

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