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  "body": "\n## Statement\n\nProvability logic abstracts away from the details of encodings used in Gödel's incompleteness theorems by expressing the provability of $\\phi$ in the given system in the language of modal logic, by means of the modality $\\Box \\phi$. That is, when $\\phi$ is a logical formula, another formula can be formed by placing a box in front of $\\phi$, and is intended to mean that $\\phi$ is provable.\n\nThen we can formalize Löb's theorem by the axiom\n\n$$\\Box(\\Box P\\rightarrow P)\\rightarrow \\Box P,$$\n\nknown as axiom GL, for Gödel–Löb. This is sometimes formalized by means of the inference rule:\n\nIf\n\n$$\\vdash \\Box P \\rightarrow P$$\n\nthen\n\n$$\\vdash P.$$\n\nThe provability logic **GL** that results from taking the modal logic **K4** (or **K**, since the axiom schema 4, $\\Box A\\rightarrow\\Box\\Box A$, then becomes redundant) and adding the above axiom GL is the most intensely investigated system in provability logic.\n",
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