proveditelné

Proveditelné is a Czech adjective meaning capable of being proven or demonstrated. In formal contexts it refers to statements, propositions, or properties for which a proof can be derived from a given set of axioms using specified inference rules. The study of such provability is central to proof theory, a branch of mathematical logic.

In a formal language L with axioms A and inference rules R, a statement φ is provable if there exists a finite sequence of formulas ending with φ where each formula is either an axiom or obtained from earlier formulas by a rule in R. Provability is relative to the chosen deductive system; the same statement may be provable in one system and not provable in another.

A key distinction in mathematics is between truth in a model and provability within a formal system. Gödel's incompleteness theorems show that in any consistent, sufficiently expressive system, there exist true statements that are not provable within that system. This demonstrates limits of provability.

Beyond pure mathematics, the term appears in philosophy and computer science. In philosophy, a claim is proved or provedatable when supported by rigorous argument and evidence. In computer science, proveditelnost is linked to formal verification: properties of algorithms or hardware can be proven correct using mathematical methods.

The concept underpins formal reasoning, mathematical rigor, and the assessment of what can be established beyond doubt within a given framework.