Counterintuitions and the well-ordering theorem
in [Logic]
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From: Aatu Koskensilta on 12 Oct 2009 14:16 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? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Nam Nguyen on 12 Oct 2009 14:29 Aatu Koskensilta 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? > Don't have a every details yet but it's possible the order be relativistic over the (more absolute) existences of the reals, which could give rise to paradoxes if we insist there's a "global" absolute well order. Imho.
From: Aatu Koskensilta on 12 Oct 2009 14:29 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Don't have a every details yet but it's possible the order be > relativistic over the (more absolute) existences of the reals, which > could give rise to paradoxes if we insist there's a "global" absolute > well order. Imho. Whatever the merits of this, it appears to be a theoretical (philosophical?) doctrine or hypothesis, not an intuition contradicted by the well-ordering theorem. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Nam Nguyen on 12 Oct 2009 15:46 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Don't have a every details yet but it's possible the order be >> relativistic over the (more absolute) existences of the reals, which >> could give rise to paradoxes if we insist there's a "global" absolute >> well order. Imho. > > Whatever the merits of this, it appears to be a theoretical > (philosophical?) doctrine or hypothesis, not an intuition contradicted > by the well-ordering theorem. > It's only as much as philosophical as the current mathematical reasoning based on the assumed knowledge of the naturals is. If you already preconceived the truth of well-order of the reals isn't philosophically based and isn' counterintuitive, well then there would be - as you said - "mysteries of sets" you'd miss. If you keep an open-minded "attitude", I'd further elaborate.
From: Rupert on 12 Oct 2009 20:21
On Oct 13, 5:16 am, 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? > Some *consequences* of the existence of a well-ordering of the reals, such as the Banach-Tarski paradox, are counter-intuitive. |