Gödel sentence ($G\leftrightarrow \neg {\mathrm{Prov}}_T(┌G┐)$)

A fixed point of the negated arithmetic provability predicate for a computably enumerable arithmetic theory $T$. Consistency excludes a proof of $G$, because a genuine proof is arithmetically verifiable and contradicts the fixed-point assertion. Omega-consistency excludes a proof of its negation. For a consistent theory extending Robinson arithmetic, this sentence is true in the standard natural numbers.

This diagonal sentence says that any proof of itself has a no-later proof of its negation. A proof of $R$ contradicts finite verification that no earlier opposite proof exists. A proof of its negation, together with finite verification that no earlier proof of $R$ exists, proves the displayed universal condition and hence $R$. Thus consistency alone excludes both proofs, strengthening the original omega-consistency form of incompleteness.