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A definable number is a real number which can be unambigously defined by some mathematical statement. Formally, one calls a real number a definable if there is some logical formula φ(x) which contains a single free variable x and such that one can prove that a is the unique real number which makes the statement φ(a) true. The formula φ(x) is not restricted to first-order statements. |
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A definable number is a real number which can be unambigously defined by some mathematical statement. Formally, one calls a real number a definable if there is some logical formula φ(x) in set theory which contains a single free variable x and such that one can prove from the Zermelo-Fraenkel-Choice set theory axioms that a is the unique real number which makes the statement φ(a) true. |
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The field of definable numbers is not complete?; there exist convergent sequences of definable numbers whose limit is not definable. However, if the sequence itself is definable in the sense that we can specify a single formula for all its terms, then its limit will be a definable number. |
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The field of definable numbers is not complete?; there exist convergent sequences of definable numbers whose limit is not definable. However, if the sequence itself is definable in the sense that we can specify a single formula for all its terms, then its limit will necessarily be a definable number. In fact, all theorems of calculus remain true if the field of real numbers is replaced by the field of definable numbers, sequences are replaced by definable sequences, sets are replaced by definable sets and functions by definable functions. |
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