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  text: '[summary: [ Gödel's second incompleteness theorem] and [ Löb's theorem] are equivalent to each other. ]\n\nThe abstract form of [ Gödel's second incompleteness theorem] states that if $P$ is a provability predicate in a [5km consistent], [-axiomatizable] theory $T$ then $T\\not\\vdash \\neg P(\\ulcorner S\\urcorner)$ for a disprovable $S$.\n\nOn the other hand, [55w Löb's theorem] says that in the same conditions and for every sentence $X$, if $T\\vdash P(\\ulcorner X\\urcorner)\\rightarrow X$, then $T\\vdash X$.\n\nIt is easy to see how GII follows from Löb's. Just take $X$ to be $\\bot$, and since $T\\vdash \\neg \\bot$ (by definition of $\\bot$), Löb's theorem tells that if $T\\vdash \\neg P(\\ulcorner \\bot\\urcorner)$ then $T\\vdash \\bot$. Since we assumed $T$ to be consistent, then the consequent is false, so we conclude that $T\\neg\\vdash \\neg P(\\ulcorner \\bot\\urcorner)$.\n\nThe rest of this article exposes how to deduce Löb's theorem from GII.\n\nSuppose that $T\\vdash P(\\ulcorner X\\urcorner)\\rightarrow X$.\n\nThen $T\\vdash \\neg X \\rightarrow \\neg P(\\ulcorner X\\urcorner)$.\n\nWhich means that $T + \\neg X\\vdash \\neg P(\\ulcorner X\\urcorner)$.\n\nFrom Gödel's second incompleteness theorem, that means that $T+\\neg X$ is inconsistent, since it proves $\\neg P(\\ulcorner X\\urcorner)$ for a disprovable $X$.\n\nSince $T$ was consistent before we introduced $\\neg X$ as an axiom, then that means that $X$ is actually a consequence of $T$. By completeness, that means that we should be able to prove $X$ from $T$'s axioms, so $T\\vdash X$ and the proof is done.',
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