Counterintuitions and the well-ordering theorem
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From: Bill Taylor on 31 Oct 2009 01:21 On Oct 29, 12:02 pm, Herman Jurjus <hjm...(a)hetnet.nl> wrote: > > I don't see why you would believe it. > > I can't say it better, at this stage, than i already did. Sorry. > > (Can you /explain/ /why/ you think the Jordan curve > should definitely be true? I can't.) Now THAT is a wonderful question, and I would dearly love to hear a response from some of the people it is directed toward! -- Wide-eyed William * Changing your mind because of emotion - that is faith. * Changing your mind because of thinking - that is philosophy. * Changing your mind because of facts - that is science.
From: Nam Nguyen on 31 Oct 2009 01:44 I'm a bit off topic here but since it's still about (counter)intuition so I'm going to ask. Let 2ZFC be a formal system written in a language L with 2 epsilon relations: L = L(e,e'). The axioms of 2ZFC are the following: - A set of axioms that has formulas using only e and all together would be a ZFC axioms (for e). - A set of axioms that has formulas using only e' and all together would be a ZFC axioms (for e'). - An additional axiom: Axy[(Au[~(uex)] /\ Av[~(ve'y)]) -> (x=y)] (This says the 2 empty sets be identical.) How would intuition perceive such a multiple epsilon relation sets? How would AC's and AD's connotations be impacted by 2ZFC?
From: Nam Nguyen on 31 Oct 2009 02:40 Nam Nguyen wrote: > I'm a bit off topic here but since it's still about > (counter)intuition so I'm going to ask. > > Let 2ZFC be a formal system written in a language > L with 2 epsilon relations: L = L(e,e'). The axioms > of 2ZFC are the following: > > - A set of axioms that has formulas using only e and > all together would be a ZFC axioms (for e). > > - A set of axioms that has formulas using only e' and > all together would be a ZFC axioms (for e'). > > - An additional axiom: > > Axy[(Au[~(uex)] /\ Av[~(ve'y)]) -> (x=y)] > > (This says the 2 empty sets be identical.) > > How would intuition perceive such a multiple epsilon relation > sets? How would AC's and AD's connotations be impacted by 2ZFC? The way my intuition is thinking is that we'd probably add more axioms to make the sets isomorphically equals between the 2 epsilons: from pairing, unions, intersection, power-sets, etc... But then when AC is required, we'd make the 2 choice functions *different* all together could be used to formulate a sort of "Axiom of Destiny" in 2ZFC. Something like that. Would you see it the same way?
From: Herman Jurjus on 31 Oct 2009 04:35 Bill Taylor wrote: > On Oct 29, 12:02 pm, Herman Jurjus <hjm...(a)hetnet.nl> wrote: > >>> I don't see why you would believe it. >> I can't say it better, at this stage, than i already did. Sorry. >> >> (Can you /explain/ /why/ you think the Jordan curve >> should definitely be true? I can't.) > > Now THAT is a wonderful question, and I would dearly love > to hear a response from some of the people it is directed toward! > > -- Wide-eyed William > > * Changing your mind because of emotion - that is faith. > * Changing your mind because of thinking - that is philosophy. > * Changing your mind because of facts - that is science. May i ask one question to you, too? What are your /reasons/ for rejecting AC, other than your acceptance of AD and the fact that ZFC+AD is inconsistent? -- Cheers, Herman Jurjus
From: Frederick Williams on 31 Oct 2009 08:46
Nam Nguyen wrote: > > I'm a bit off topic here but since it's still about > (counter)intuition so I'm going to ask. > > Let 2ZFC be a formal system written in a language > L with 2 epsilon relations: L = L(e,e'). Assuming that e and e' have intended interpretations, what are they? > The axioms > of 2ZFC are the following: > > - A set of axioms that has formulas using only e and > all together would be a ZFC axioms (for e). > > - A set of axioms that has formulas using only e' and > all together would be a ZFC axioms (for e'). > > - An additional axiom: > > Axy[(Au[~(uex)] /\ Av[~(ve'y)]) -> (x=y)] > > (This says the 2 empty sets be identical.) > > How would intuition perceive such a multiple epsilon relation > sets? How would AC's and AD's connotations be impacted by 2ZFC? -- Which of the seven heavens / Was responsible her smile / Wouldn't be sure but attested / That, whoever it was, a god / Worth kneeling-to for a while / Had tabernacled and rested. |