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  "body": "\n## Summary\n\nA complete theory is a satisfiable [[theory|92SWXST1ZM]] that decides every sentence of its language: it implies the sentence or its negation.\nAll models of a complete theory are [[elementarily equivalent|P3B1V182N4]].\nThe theory of a single structure is always complete, and proving that a list of axioms is complete is a central task of model theory.\n\n## Explanations\n\n### L1 human\n\nA complete theory leaves no first-order question open.\nThe axioms of groups are not complete: they do not decide whether $x \\cdot y = y \\cdot x$ holds for all $x$ and $y$.\nThe axioms of dense linear orders without endpoints are complete: every first-order question about such an order has the same answer in $(\\Q, \u003c)$ and in $(\\R, \u003c)$.\n",
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