Can integers be defined in the first-order theory of the rationals with addition, multiplication, and order?
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1$\begingroup$ Order is unnecessary, it is definable from addition and multiplication. $\endgroup$– André NicolasFeb 10, 2014 at 7:12
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$\begingroup$ True, I and a friend had just worked this out. Somehow I found it simpler to state it this way. $\endgroup$– PrateekFeb 10, 2014 at 7:13
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1 Answer
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Yes. It's a much celebrated theorem of Julia Robinson.
Julia Robinson, Definability and Decision Problems in Arithmetic. The Journal of Symbolic Logic, Vol. 14, No. 2 (Jun., 1949) , pp. 98-114
answered Feb 10, 2014 at 6:51