SCI.MATH POLL - uncountable infinity
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From: herbzet on 5 Jun 2010 09:14 |-|ercules wrote: > "herbzet" wrote > > herbzet wrote: > >> |-|ercules wrote: > >> > >> > I'll wait and see if someone else takes the bait. > >> > > >> > >> The proof of higher infinities than 1,2,3...oo infinity relies on > >> > >> the fact that there is no box that contains all and only all the > >> > >> label numbers of the boxes that don't contain their own label number. > >> > > >> > TRUE OR FALSE > >> > >> Um, false, so far as I know. > >> > >> We have > >> > >> 1) |N| < |P(N)| > >> 2) |P(N)| <= |R| > >> -------------- > >> .: |N| < |R| > >> > >> but neither of Cantor's proofs that |N| < |R| involves either of > >> premises (1) or (2), as far as I can recall. > > > > Why do you ask? > > > > Because the most widely used proof of uncountable infinity is the > contradiction of a bijection from N to P(N), which is analagous to > the missing box question. I'm not sure that this proof is really a "proof of uncountable infinity" anyway. A finitist, for example, would reject the notion that the naturals constitute an infinite set in the first place, but I see no reason why she would reject the proof that for any set S, |S| < |P(S)|. -- hz
From: herbzet on 5 Jun 2010 09:37 |-|ercules wrote: > "herbzet" wrote ... > > |-|ercules wrote: > >> herbzet wrote: > >> > herbzet wrote: > >> >> |-|ercules wrote: > >> >> > >> >> > I'll wait and see if someone else takes the bait. > >> >> > > >> >> > >> The proof of higher infinities than 1,2,3...oo infinity relies on > >> >> > >> the fact that there is no box that contains all and only all the > >> >> > >> label numbers of the boxes that don't contain their own label number. > >> >> > > >> >> > TRUE OR FALSE > >> >> > >> >> Um, false, so far as I know. > >> >> > >> >> We have > >> >> > >> >> 1) |N| < |P(N)| > >> >> 2) |P(N)| <= |R| > >> >> -------------- > >> >> .: |N| < |R| > >> >> > >> >> but neither of Cantor's proofs that |N| < |R| involves either of > >> >> premises (1) or (2), as far as I can recall. > >> > > >> > Why do you ask? > >> > > >> > >> Because the most widely used proof of uncountable infinity is the > >> contradiction of a bijection from N to P(N), which is analagous to > >> the missing box question. > > > > Perhaps so, but why do you ask? > > It's hard to explain to you when you answer FALSE and then PERHAPS to identical > questions. > Anyway, I'm going to sleep [...] I think you need some sleep -- I didn't answer "perhaps" to any question, much less a question identical to your poll question. Just to clear up any ambiguity, my "perhaps" was directed to your assertion that the most widely used proof of uncountable infinity is the contradiction of a bijection from N to P(N) -- which I doubt -- not to your subordinate assertion that the proof is analagous to the missing box question, an assertion whose truth I freely grant. -- hz
From: herbzet on 5 Jun 2010 09:43 herbzet wrote: > Just to clear up any ambiguity, my "perhaps" was directed to your > assertion that the most widely used proof of uncountable infinity > is the contradiction of a bijection from N to P(N) -- which I doubt -- > [...] That is, I doubt your assertion, not the proof. English is funny, no? -- hz
From: herbzet on 5 Jun 2010 10:41 |-|ercules wrote: > "herbzet" wrote ... > > |-|ercules wrote: > >> herbzet wrote: > >> > herbzet wrote: > >> >> |-|ercules wrote: > >> >> > >> >> > I'll wait and see if someone else takes the bait. > >> >> > > >> >> > >> The proof of higher infinities than 1,2,3...oo infinity relies on > >> >> > >> the fact that there is no box that contains all and only all the > >> >> > >> label numbers of the boxes that don't contain their own label number. > >> >> > > >> >> > TRUE OR FALSE > >> >> > >> >> Um, false, so far as I know. > >> >> > >> >> We have > >> >> > >> >> 1) |N| < |P(N)| > >> >> 2) |P(N)| <= |R| > >> >> -------------- > >> >> .: |N| < |R| > >> >> > >> >> but neither of Cantor's proofs that |N| < |R| involves either of > >> >> premises (1) or (2), as far as I can recall. > >> > > >> > Why do you ask? > >> > > >> > >> Because the most widely used proof of uncountable infinity is the > >> contradiction of a bijection from N to P(N), which is analagous to > >> the missing box question. > > > > Perhaps so, but why do you ask? > > It's hard to explain to you when you answer FALSE and then PERHAPS to identical questions. > > Anyway, I'm going to sleep and aren't posting any more, May flights of angels SING thee to thy rest! Hurry back, sweet Herkimer. -- hz
From: George Greene on 5 Jun 2010 17:12
On Jun 5, 9:14 am, herbzet <herb...(a)gmail.com> wrote: > I'm not sure that this proof is really a "proof of uncountable infinity" > anyway. A finitist, for example, would reject the notion that the naturals > constitute an infinite set in the first place, but I see no reason > why she would reject the proof that for any set S, |S| < |P(S)|. Please don't say "a finitist". There is no such thing. There is no finite number of the finite things. Therefore, even if you want to contemplate only finite things, there will still NOT be only finitely many things to contemplate. Finitism is simply not a coherent position. |