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Someone said that Dedekind had a "theory of real numbers", and I wonder about the truth level of that statement. He published the well-known construction of real numbers in 1872, partly inspired by earlier ideas, just like how things usually work out. Today it is treated as an aside, certainly in textbooks on real analysis, it might be mentioned and presented in an appendix, just to show that a set of real numbers does exist as a structure in set theory. I have never before seen mention of an entire theory developed by Dedekind, which if it exists would predate Tarski's formal treatment of the theory of the real ordered field, or something less, or not at all? At the time, maybe Dedekind's paper got considered the basis of a theory, but that just faded away?

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    $\begingroup$ I guess it depends on what you mean by "theory." $\endgroup$ – saulspatz Jul 29 '18 at 18:03
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    $\begingroup$ Dedekind cuts (or sections) as explained in the comments here $\endgroup$ – rtybase Jul 29 '18 at 18:20
  • $\begingroup$ I got taught mathematics so that "theory" means something with axioms in it, plus whatever statements can be derived from them as theorems. Either Dedekind did formulate axioms, or he did something else that got to become considered a "theory" anyways. Or what I heard is probably just BS. $\endgroup$ – Tommy R. Jensen Jul 30 '18 at 12:51

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