On Wednesday, 19 June 2013 14:12:36 UTC+2, Peter Percival wrote: > email@example.com wrote: > > Look: Zermelo has "proved" that the real numbers can be well-ordered. But it has been proved that the real numbers in fact cannot be well-ordered. That is as certain as that you cannot get a 13 with two dice. I think you're confusing two different results.
But your thinking is mistaken. It is obvious that the reals have not yet been well-ordered. If there had been a positive result, everybody would know it.
Further there would be a contradiction: In order to well-order elements, you must be able to distinguish them. But it is impossible to distinguish more than countably many.
The axiom of choice leads to the well-ordering theorem. Zermelo proved "every set can be well-ordered", which is literally a lie. We know that it is impossible to well-order any uncountable set because there are only countably many marks which can be attached to the elements. That much is not sufficient to well order even the smallest uncountable set, let alone a set of, say, cardinal number aleph_3. Without AC (which for any real attempt of well-ordering does help as much as dice for playing chess) we have:
"For example, it is a theorem that there does not exist any way to ever actually construct or even define a well-ordering of the real numbers." [Bill Thurston: "On proof and progress in mathematics", Bull. of the American Math. Soc. 30, 2, 1994, pp. 161-177]
"Feferman and Levy showed that one cannot prove that there is any non- denumerable set of real numbers which can be well ordered." [Abraham A. Fraenkel, Yehoshua Bar-Hillel, Azriel Levy: "Foundations of Set Theory", North Holland, Amsterdam (1973) p. 62]
So it is clear that no sensible mathematician would bet 1 cent on the announcement of a well-ordering of the reals until 2020, say. An axiom directly stating the contrary would easily be recognized as counterfactual. But by Zermelo's "proof" many freshmen get the false impression that Zermelo had something proven that really could be done. It cannot.
> There are models of ZFC in which a well-ordering is a not definable.
There is no model of ZFC in which a well-ordering has ever been constructed.
> That does not mean that R cannot be well ordered, it just means that if phi(x,y) is the ordering relation, phi is not a formula of ZFC.
That is not of interest. Of interest is this: formulas of ZFC are nothing anybody can depend on.