Obections to Cantor's Theory (Wikipedia article)
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From: David Kastrup on 25 Jul 2005 18:53 malbrain(a)yahoo.com writes: > David Kastrup wrote: >> Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: >> >> > This is precisely the kind of question which cannot be >> > answered. There is no bound to the lengths of the strings, but you >> > claim they are finite. >> >> Can you tell us the difference between "arbitrarily large" and >> "infinite"? > > Leading the discussion into a circle won't help. karl m Well, that difference is the fundamental problem of Tony as far as I can see. So getting him to realize the difference is the only way I see to get _out_ of the circle. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: malbrain on 25 Jul 2005 19:02 David Kastrup wrote: > malbrain(a)yahoo.com writes: > > > David Kastrup wrote: > >> Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > >> > >> > This is precisely the kind of question which cannot be > >> > answered. There is no bound to the lengths of the strings, but you > >> > claim they are finite. > >> > >> Can you tell us the difference between "arbitrarily large" and > >> "infinite"? > > > > Leading the discussion into a circle won't help. karl m > > Well, that difference is the fundamental problem of Tony as far as I > can see. So getting him to realize the difference is the only way I > see to get _out_ of the circle. Tony was asked to provide a number for the count of elements in the set N. He said it was impossible. That's an improvement. Your question doesn't lead anywhere. karl m
From: Virgil on 25 Jul 2005 19:16 In article <MPG.1d4f082a49cdb89d989f7b(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Virgil said: > > But there is no 1-1 correspondence between naturals and INFINITE > > bit strings (only with finite bit strings). This is just another > > instance of that same delusion that TO has that there exist > > naturals with more than finitely many naturals as predecessors. > > > If they have all 1's in finite positions, then there is indeed a > bijection between infinite bit strings and finite naturals. Not in any standard representation, in which the leading digit of an n-ary representation of a natural must be non-zero. > If they have any 1's in positions infinitely to the left of the > point, then they represent sets that include infinite integers, and > also have infinite values as binary numbers. If one includes all such strings of infinitely many 0's and 1's, there are as many such strings as there are subsets of N, which is a cardinality greater than that of N itself. > The bijection works perfectly, IF one allows infinite integers. Since no sane person will allow infinite NATURALS, TO's infinite integers are exclused from N, by everyone except himself. > This > restriction on the whole numbers is the only thing standing in the > way of bijection betweem [0,1) and N. So that TO suggests we make over N by inserting as many extra members as needed to biject with [0,1)?
From: Daryl McCullough on 25 Jul 2005 19:06 Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > >Dik T. Winter said: >> 1 = 1 >> 2 = 2 >> 3 = 11 >> 4 = 12 >> 5 = 21 >> 6 = 22 >> 7 = 111 >> 8 = 112 >> 9 = 121 >> 10 = 122 >> etc. >> In this representation each finite natural number is represented by a single >> finite string. Now how many of such finite strings are there, given that the >> stringlength is unbounded? >This is precisely the kind of question which cannot be answered. It can be answered by anyone competent. The answer is "aleph-0", which is the smallest infinite cardinality. >There is no bound to the lengths of the strings, but you claim they >are finite. What number am I supposed to use to calculate this set size? There is no "number" that can be the set size. The set size is aleph-0, (also called "omega") which is not a (natural) number. >Still, the question is interesting if infinite digits are allowed. If you allow infinitely many digits, then the cardinality is the same as that of the reals, namely 2^omega. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 25 Jul 2005 19:14
Tony Orlow says... > >Martin Shobe said: >> Yep. But be careful here, at *every* stage of this process, we have >> still only done it a finite number of times. > >Uhhh.... Look at what you just agreed to. The number of times you are adding 1 >is infinite. Fact 1: To create any *specific* natural number only requires finitely many applications of the successor operator. Fact 2: To create the entire set of *all* natural numbers requires infinitely many applications of the successor operator. Finitely many for the first natural, plus finitely many for the second, plus finitely many for the third, etc. -- Daryl McCullough Ithaca, NY |