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  "body": "\n## Summary\n\nA Hilbert system is a proof system with many axiom schemas and very few rules of inference, often only modus ponens.\nA proof is a finite list of formulas in which each formula is an axiom, a hypothesis, or follows from earlier ones by a rule.\nHilbert systems are simple to describe and to reason about, which makes them the standard tool for results about provability.\nThey are named after David Hilbert.\n\n## Explanations\n\n### L1 human\n\nIn a Hilbert system, almost all the work is done by the axioms.\nThe only move is modus ponens: from $\\varphi$ and $\\varphi \\to \\psi$, write down $\\psi$.\nA proof is then just a numbered list of lines, each justified by an axiom or by modus ponens.\nFor example, the formula $p \\to p$ has a proof of five lines in the usual system.\nSuch proofs are easy to check but often hard to find, which is why [[natural deduction|6KT3BJZ7D1]] was invented.\n",
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