Obections to Cantor's Theory (Wikipedia article)
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From: Tony Orlow on 27 Jul 2005 17:10 Robert Low said: > Tony Orlow (aeo6) wrote: > > Robert Low said: > >>Are there rationals in [0,1) where p and/or q have to be infinite? > > Interesting question. In order to have an infinite number of them, yes, you > > would need either an infinite number of numerators or an infinite number of > > denominators, both of which are whole numbers. So, you would require infinite > > values in either numerator or denominator, in order to achieve an infinite set. > > OK, then, so now I consider the ratio 314159.../100000000000, where both > are 'infinite integers'. (I'm clearly playing the game of 'it looks > a bit like' where I ought to be playing the game of 'I can prove'; > but that's for obvious reasons.) It looks to me as if their ratio is > pi, which is not rational. It looks close to pi. Now all you have to do is specify the infinite number of other digits, and you're there! Good luck! > > Anyway, I'd still like to know how many elements there are in > the set of all finite integers. When you can answer that, there > might be some scope for further development. > LOL. You call it aleph_0, so why should I argue? It's no wonder you are satisfied with set sizes that you can't do anything with. If that's the size, though, it's a finite number. I suppose it's a wonderful name for a number that can't be pinned down. -- Smiles, Tony
From: David Kastrup on 27 Jul 2005 17:12 stevendaryl3016(a)yahoo.com (Daryl McCullough) writes: > Tony Orlow <aeo6(a)cornell.edu> said: > >>So, the sum of a finite number of finite terms is infinite. Sure. > > No. The sum of a finite number of finite terms is finite, and the > sum of an infinite number of finite terms is infinite. Last time I looked, there was no such thing as a sum of an infinite number of terms (since normal sum properties like associativity and commutativity fail in the infinite). Even if it were, like assuming for the sake of argument something like infty ---- \ -k > 2 / ---- k = 0 was a proper sum (instead of a limit), then the most reasonable interpretation of the sum of this infinite number of finite terms most certainly would not be infinite. It would be 2. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Robert Low on 27 Jul 2005 17:13 Tony Orlow (aeo6) wrote: > Robert Low said: >>Tony Orlow (aeo6) wrote: >>>Robert Low said: >>>>Are there rationals in [0,1) where p and/or q have to be infinite? >>>Interesting question. In order to have an infinite number of them, yes, you >>>would need either an infinite number of numerators or an infinite number of >>>denominators, both of which are whole numbers. So, you would require infinite >>>values in either numerator or denominator, in order to achieve an infinite set. >>OK, then, so now I consider the ratio 314159.../100000000000, where both >>are 'infinite integers'. (I'm clearly playing the game of 'it looks >>a bit like' where I ought to be playing the game of 'I can prove'; >>but that's for obvious reasons.) It looks to me as if their ratio is >>pi, which is not rational. > It looks close to pi. Now all you have to do is specify the infinite number of > other digits, and you're there! Good luck! That's dead easy. They're the remaining digits in the decimal expansion of pi. Not only is that sequence well-defined, it's even computable. So what is the difference between a rational number and an irrational one? >>Anyway, I'd still like to know how many elements there are in >>the set of all finite integers. When you can answer that, there >>might be some scope for further development. > LOL. You call it aleph_0, so why should I argue? You already do argue. You claim it is a finite number. aleph-O isn't a one of them. Weaseling isn't an anwer any more than making it up as you go along without contemplating the consequences of your next blithe statement.
From: Tony Orlow on 27 Jul 2005 17:15 Daryl McCullough said: > Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > > > >Daryl McCullough said: > > >> Why do you keep saying that? It's provably false. The set of all > >> finite strings is an infinite set. It's infinite by *your* definition > >> of infinite, in the sense that it is "without end". The set of all > >> finite strings is the union of > >> > >> S_1 = the set of strings of length 1 > >> S_2 = the set of strings of length 2 > >> S_3 = the set of strings of length 3 > >> ... > >> > >> The collection of subsets S_n goes on without end. > >So, each of these sets is finite right, given finite S and L? > > Right. For each n, S_n is finite. > > >There are an infinite number of such finite sets? > > Yes, there are infinitely many possible values for n. > > >Do they then go, say, from S_1 to S_oo? > > The sequence S_1, S_2, ... is an *infinite* sequence; > it's a sequence that has no end; it has no last set. > So no, the last set is not S_oo because there is no > last set. But there are an infinite number of them, so your subscript must become infinite at some point, or are you repeating finite subscripts? > > >And S_1 is the set of strings of length 1, and S_2 > >is the set of strings of length 2, etc, so S_n is > >the set of strings of length n? Okay. What length > >are the strings in S_oo? > > There *is* no S_oo. The possible values of n are the *finite* > naturals. But there are an infinite number of them, right? If there were a million, then the last would be S_million. If there were a googol, then the last set would be S_googol. You may say there is no last set, but if there are an infinite number before ANY given set, then the subscript for that set must be infinite. If not, then there is no set with an infinite number of preceding sets, and there is not an infinite number of sets. > > I'm telling you that S_1, S_2, ... is a sequence without end, > with no last set, and you are asking me what the length > of the strings in the last set. That doesn't make any sense. > > There is no last set. The sequence goes on forever. And when you have an infinite number of them, you still have finite subscripts? How is that possible? > > -- > Daryl McCullough > Ithaca, NY > > -- Smiles, Tony
From: Tony Orlow on 27 Jul 2005 17:16
Daryl McCullough said: > Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > > >If you place no restriction on the length of strings, then they can be > >infinitely long. > > No, there is only one restriction on the length of strings, and that > is that the length is finite. > > -- > Daryl McCullough > Ithaca, NY > > Oh, well you had said there was NO restiction, and they could get arbitrarily large. I guess that wasn't true. -- Smiles, Tony |