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  "body": "\n## Statement\n\nLet $\\sigma$ be a signature, let $\\Gamma$ be a set of first-order [[sentences|DKFGQTRYZE]] over $\\sigma$, and let $\\varphi$ be a sentence over $\\sigma$.\nLet $\\vdash$ be derivability in a standard [[Hilbert system|1MGMQX43B8]] for first-order logic with equality.\nIf $\\Gamma \\models \\varphi$, as a [[logical consequence|1HK0VYWTV7]], then $\\Gamma \\vdash \\varphi$.\n\n## Remarks\n\nTogether with [[soundness|4A84VHDTHV]], the theorem gives that $\\Gamma \\vdash \\varphi$ if and only if $\\Gamma \\models \\varphi$.\nAn equivalent form: every consistent set of sentences has a [[model|MVCH6N9SNW]].\nThe usual proof is Henkin's.\nExtend a consistent $\\Gamma$ to a maximal consistent set with witnesses: for each sentence $\\exists x \\, \\chi$ in it, a constant $c$ with $\\chi(c)$.\nThe closed terms, modulo provable equality, then form a model.\nSince derivations are finite, the theorem implies the [[compactness theorem|1NR3HAWG9D]].\nFor example, every sentence true in all groups is derivable from the group axioms.\nThe theorem is about first-order logic; full second-order logic has no sound and complete proof system of this kind.\nKurt Gödel proved the theorem in his doctoral thesis of 1929.\n\n## Explanations\n\n### L1 human\n\nThe theorem says that the rules of first-order proof are strong enough.\nWhatever follows from some axioms, in the sense that it is true in every situation where the axioms are true, can actually be proved from them.\nSo \"true in all models\" and \"provable\" mean the same thing.\n\n### L2 human\n\nThe completeness theorem links syntax with semantics.\nTo show that something is unprovable, it is enough to build a model where the axioms hold and the statement fails.\nThis is how independence results, such as those in set theory, are proved.\nIt should not be confused with the [[incompleteness theorems|X9WYNYE29C]]: these say that some theories are incomplete, meaning that they neither prove nor refute some sentence.\nBoth results hold at once, since such a sentence is true in some models of the theory and false in others.\n",
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