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  "body": "\n## Summary\n\nA theory has quantifier elimination when every formula is equivalent, modulo the theory, to a formula without quantifiers.\nThen the definable sets of its models are simple to describe, and questions about the theory reduce to questions about quantifier-free formulas.\nQuantifier elimination often gives completeness and decidability.\nClassical examples are dense linear orders without endpoints, [[algebraically closed fields|MT8TDVPBXB]], and [[real closed fields|DDY3Q3PJ6W]] in the language of ordered rings.\n\n## Explanations\n\n### L1 human\n\nQuantifiers are the words \"for all\" and \"there exists\".\nA theory has quantifier elimination if every statement that uses them can be rewritten as a statement that does not use them.\nThe rewriting must keep the meaning in every model of the theory.\nFor example, over the real numbers, \"there exists $x$ with $x^2 + bx + c = 0$\" means the same as \"$b^2 - 4c \\ge 0$\".\nAfter the rewriting, one only has to check equations and inequalities about the given numbers.\n",
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