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  "body": "\n## Summary\n\nProvability logic is the [[modal logic|C3RZY20ETK]] in which $\\square\\varphi$ is read as \"$\\varphi$ is provable\" in a formal theory such as Peano arithmetic.\nIts main system $\\mathrm{GL}$ is the normal modal logic given by Löb's axiom $\\square(\\square p \\to p) \\to \\square p$.\nBy Solovay's theorem, $\\mathrm{GL}$ proves exactly the modal laws of provability in Peano arithmetic, so questions about provability become finite combinatorics of Kripke frames.\n\n## Explanations\n\n### L1 human\n\nGödel showed that a formal theory can talk about its own proofs.\nProvability logic collects the general laws of \"it is provable that\" into a small modal system.\nOne law is \"if $p$ is provable, then it is provable that $p$ is provable\", the formula $\\square p \\to \\square\\square p$.\n\"If $p$ is provable, then $p$\" is not a law of the system.\nWith $p$ the false sentence $0 = 1$, it says that the theory is consistent, and by Gödel's second incompleteness theorem the theory cannot prove this.\n",
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