= Equivalence of a Gödel sentence and a consistency statement
{c}
{title2=$T\vdash G\leftrightarrow\operatorname{Con}(T)$}
For a <Gödel sentence> $G\leftrightarrow\neg\Box G$ with the <Hilbert-Bernays-Löb derivability conditions>, fix a refutable contradiction sentence $\bot$. Since $\bot\to G$ is provable, $\Box\bot\to\Box G$, giving $G\to\neg\Box\bot$. For the reverse direction, $\Box G$ implies both $\Box\Box G$ and $\Box\neg\Box G$, hence $\Box\bot$. Contraposition gives $\neg\Box\bot\to\neg\Box G\to G$. This is an internal equivalence of sentences, not a rule asserting the theory's own consistency.
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