Counterintuitions and the well-ordering theorem
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From: Bill Taylor on 2 Dec 2009 23:45 Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > It is often said that the well-ordering theorem, or the existence of a > well-ordering of the reals in particular, is counterintuitive. Alas, > I've never quite fathomed what intuitions are contradicted. Perhaps > someone with keener intuition into the mysteries of sets can shed some > light on this pressing matter? Well, now the thread has died down a bit, I take it that this intital query has been satisfactorily answered? -- Wondering Willy
From: WM on 4 Dec 2009 01:55
On 3 Dez., 05:45, Bill Taylor <w.tay...(a)math.canterbury.ac.nz> wrote: > Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > It is often said that the well-ordering theorem, or the existence of a > > well-ordering of the reals in particular, is counterintuitive. Alas, > > I've never quite fathomed what intuitions are contradicted. Perhaps > > someone with keener intuition into the mysteries of sets can shed some > > light on this pressing matter? > > Well, now the thread has died down a bit, I take it that this > intital query has been satisfactorily answered? Yes. The existence of a well-ordering of the reals is not counterintuitive but it is simply wrong. Counterintuitive is only that there are mathematicians who have not yet recognized that. Zermelo proved in 1904: "If every set can be well ordered then every set can be well ordered." But unfortunately he missed the fact that he proved only the assumption. If you look into his "proof" with this information you should easily see it. Regards, WM |