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  "body": "\n## Statement\n\nThe completeness theorem can also be understood in terms of consistency, as a consequence of Henkin's model existence theorem. We say that a theory *T* is *syntactically consistent* if there is no sentence *s* such that both *s* and its negation ¬*s* are provable from *T* in our deductive system. The model existence theorem says that for any first-order theory *T* with a well-orderable language,\n\nif $T$ is syntactically consistent, then $T$ has a model.\n\nAnother version, with connections to the Löwenheim–Skolem theorem, says:\n\nEvery syntactically consistent, countable first-order theory has a finite or countable model.\n\nGiven Henkin's theorem, the completeness theorem can be proved as follows: If $T \\models s$, then $T\\cup\\lnot s$ does not have models. By the contrapositive of Henkin's theorem, then $T\\cup\\lnot s$ is syntactically inconsistent. So a contradiction ($\\bot$) is provable from $T\\cup\\lnot s$ in the deductive system. Hence $(T\\cup\\lnot s) \\vdash \\bot$, and then by the properties of the deductive system, $T\\vdash s$.\n",
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