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

For a computably enumerable first-order theory $T$ extending Robinson arithmetic, choose the standard arithmetic arithmetical provability predicate ${\mathrm{Prov}}_T$. The diagonal lemma, carried out in the weak base theory, gives a sentence $G$ withIf $T$ is consistent, $T$ cannot prove $G$: an actual finite proof would be verifiable in the weak base, so $T$ would prove both its provability and its unprovability. There is therefore no actual proof of $G$, and the displayed equivalence, true in the standard natural numbers, shows that this canonical $G$ is true there.

If $T$ is omega-consistent, it cannot prove $\neg G$ either. Such a proof would imply the existence of a proof code for $G$, while each standard numeral can be individually verified not to be such a code. Ordinary consistency suffices for unprovability of $G$; the original argument for unprovability of its negation uses the stronger hypothesis. Requiring the fixed-point equivalence in a sound weak base is important: an arbitrary equivalence provable only in an unsound extension need not yield a true sentence.

Assuming PA is consistent, the theory $S=\mathrm{PA}+\neg {\mathrm{Con}}_{\mathrm{PA}}$ is consistent by second incompleteness. Its false axiom $G=\neg {\mathrm{Con}}_{\mathrm{PA}}$ is provable, so the standard semantic biconditional $G\leftrightarrow \neg {\mathrm{Pr}}_S(┌G┐)$ is true with both sides false. Thus a true biconditional alone does not make an arbitrary sentence a true canonical Gödel sentence. A canonical Pi-1 construction adds the necessary syntactic or sound-base information. Truth, internal equivalence and canonical construction must be distinguished.

For a Gödel sentence $G\leftrightarrow \neg \square G$ with the Hilbert-Bernays-Löb derivability conditions, fix a refutable contradiction sentence $\perp$. Since $\perp \to G$ is provable, $\square \perp \to \square G$, giving $G\to \neg \square \perp$. For the reverse direction, $\square G$ implies both $\square \square G$ and $\square \neg \square G$, hence $\square \perp$. Contraposition gives $\neg \square \perp \to \neg \square G\to G$. This is an internal equivalence of sentences, not a rule asserting the theory's own consistency.

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.

For disjoint sets numeralwise enumerated by $P,R$, finite-initial-segment and total-comparison axioms let $H$ separate the first set from the second in the theory. For a member of the first set use a positive $P$ stage and finitely many refutations of $R$. For a member of the second set use its positive $R$ stage: earlier proposed $P$ stages are all refuted by a finite bounded proof, while later stages lose to that known $R$ witness. This proves a separator without inferring arbitrary negative facts from the mere Sigma-1 form of the enumerators.

For the standard Gödel sentence and Rosser sentence, the formalized Rosser unprovability theorem and $G\leftrightarrow \mathrm{Con}(T)$ yield $T⊢G\to R$. If $T+\mathrm{Con}(T)$ is consistent, the converse is unprovable under the usual formalized incompleteness hypotheses. A proof of $R\to G$ would give $\square \neg G\to \square \neg R$. Formalized negative unprovability identifies $\neg \square \neg R$ with $\mathrm{Con}(T)$ and $\neg \square \neg G$ with $\mathrm{Con}(T+\mathrm{Con}(T))$. The strengthened theory would then prove its own consistency, contradicting the Gödel second incompleteness theorem. The consistency assumption belongs to this strictness conclusion.