ofullständighet

Ofullständighet, or incompleteness, is a concept used in logic, mathematics and philosophy to describe the lack of a system's ability to settle all questions expressible within its framework. In simple terms, a theory is incomplete when there are statements that are true but cannot be proven using the theory's axioms, or statements whose truth cannot be determined from them.

In mathematical logic, the most famous results are Gödel's incompleteness theorems from 1931. They apply to any formal, effectively axiomatized theory that is rich enough to formalize basic arithmetic (for example Peano arithmetic, or ZFC set theory). The first theorem states that such a theory cannot be both consistent and complete: there exist true statements about natural numbers that are unprovable within the theory. The second theorem shows that the theory cannot prove its own consistency from inside. These results do not deny the truth of unprovable statements; they reveal intrinsic limits in the formal approach to foundational mathematics.

Outside pure logic, incompleteness appears in computer science (undecidability and the halting problem), epistemology (limits of knowledge), and philosophy of mathematics (the status of mathematical truth). Some researchers address incompleteness by adopting new axioms or alternate foundations, acknowledging that no single formal system may capture all mathematical truths.