Decidability of first-order theories of the real numbers

In mathematical logic, a first-order language of the real numbers is the set of all well-formed sentences of first-order logic that involve universal and existential quantifiers and logical combinations of equalities and inequalities of expressions over real variables. The corresponding first-order theory is the set of sentences that are actually true of the real numbers. There are several different such theories, with different expressive power, depending on the primitive operations that are allowed to be used in the expression. A fundamental question in the study of these theories is whether they are decidable: that is, whether there is an algorithm that can take a sentence as input and produce as output an answer "yes" or "no" to the question of whether the sentence is true in the theory.

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