Definition
Omega-incompleteness (or ω-incompleteness) is a property of a formal system where, for some predicate P, the system can prove P(0), P(1), P(2), ... — every specific numeric instance — but cannot prove the universal ∀n: P(n). A related notion, ω-inconsistency, is when the system proves every P(n) and proves ¬∀n: P(n), which is technically consistent (no outright contradiction) but semantically broken.
Why it matters
How it works
To diagnose ω-incompleteness in a system, look for a predicate P such that every instance can be derived (often via induction-free direct computation) but the universal ∀n: P(n) requires a strictly stronger inductive principle the system lacks. The classic example involves the Gödel sentence and related constructs. ω-consistency rules out the pathological case where ¬∀n: P(n) is also provable; ordinary consistency does not.