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: Transfer Principle on 6 Jun 2010 02:24
On Jun 4, 12:04 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 4, 1:37 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > How about this: ZFC proves _neither_ or _both_ of > > CH and its negation (Goedel and Cohen)? > I see. So do you have any confidence that ZF is consistent? Here is my opinion on this issue. I am confident that ZF is consistent, because I believe that if ZF were inconsistent, a proof of this would have been found by now. The rapidity of how a proof of the inconsistency of naive set theory was found does suggest that a proof of ~Con(ZF) would've been found equally quickly, if such a proof were possible. If it does turn out that ZF is inconsistent, then there would have been some underlying reason that the proof wasn't discovered for over a century after the axioms were first given. This is why I often refer to the mathematician Ed Nelson, who's currently working on a proof that PA (and hence ZF) is inconsistent. Nelson's work depends on very large natural numbers -- naturals so large that one has to come up with a new operation, called tetration, to describe them. Tetration is an operation that isn't considered by most mathematicians, and so a proof which relies on tetration can easily be overlooked for more than a century. The fact that the consistency strength of PA (Gentzen) is epsilon_0, the smallest ordinal which can't be constructed from omega with finitely many additions, multiplications, and exponentiations, but can be written with finitely many _tetrations_ -- indeed, it is omega^^omega (Rucker) -- is suspicious. It indicates to me that the key to a potential proof of ~Con(PA) (hence ~Con(ZF)) is the operation of tetration. But that being said, even though it's unlikely that we'll ever see a proof that ZF is inconsistent (even from Nelson), this doesn't mean that I must a priori agree that Herc is wrong. Although in Cooper's polls, I will vote for the choice which favors ZFC (and thus Herc opposes), the other choice isn't "wrong." There can be theories other than ZFC in which the other choice in the poll is right. Similarly, I'll vote for "0.999... = 1" in a poll asking for whether these two values are equal, but I'll still consider theories which prove that they are distinct. To conclude, just because I believe that ZF is consistent, it doesn't mean I believe that Cooper must be wrong. |