An uncountable countable set
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From: Tony Orlow on 7 Oct 2006 23:45 mueckenh(a)rz.fh-augsburg.de wrote: > Tony Orlow schrieb: > >> mueckenh(a)rz.fh-augsburg.de wrote: >>> Tony Orlow schrieb: >>> >>> >>>>>> Why not? Each and every number of the list terminates. That one is a number >>>>>> that does *not* terminate. >>>>>> >>>>>> > If you think that 0.111... is a number, but not in the list, >>>>> It is me who insists that it is not a representation of a number. >>>> Well, Wolfgang, that sets us apart, though I agree it's not a "specific" >>>> number. It's still some kind of quantitative expression, even if it's >>>> unbounded. Would you agree that ...333>...111, given a digital number >>>> system where 3>1? >>> That is the similar to 0.333... > 0.111.... But all these >>> representations exist only potentially, in my opinion. The difference >>> is, that 0.333... can be shown to lie between two existing numbers, so >>> we can calculate with it, while for ...333 this cannot be shown. >> I think it can be shown to lie between ...111 and ...555, given that >> each digit is greater than the corresponding digit in the first, and >> less than the corresponding digit in the second. > > Yes, but only if we define, for instance, > > A n eps |N : 111...1 < 333...3 where n digits are symbolized in both > cases. > > This approach would be comparable with the "measure" which gives > > A n eps |N : |{1,2,3,...,2n}| = 2*|{2,4,6,...,2n}|. > > I don't know whether these definitions are of any use, but I am sure > that they are not less useful than Cantor's cardinality. > > Regards, WM > > . > My opinion about that is, if one wants to talk about what happens "at infinity", that's the way that makes sense, not the measureless way of abstract set theory. I trust limit concepts, but not limit ordinals. Tony
From: Tony Orlow on 7 Oct 2006 23:50 mueckenh(a)rz.fh-augsburg.de wrote: > Tony Orlow schrieb: > >> Han de Bruijn wrote: >>> stephen(a)nomail.com wrote: >>> >>>> Han.deBruijn(a)dto.tudelft.nl wrote: >>>> >>>>> Worse. I have fundamentally changed the mathematics. Such that it shall >>>>> no longer claim to have the "right" answer to an ill posed question. >>>> Changed the mathematics? What does that mean? >>>> The mathematics used in the balls and vase problem >>>> is trivial. Each ball is put into the vase at a specific >>>> time before noon, and each ball is removed from the vase at >>>> a specific time before noon. Pick any arbitrary ball, >>>> and we know exactly when it was added, and exactly when it >>>> was removed, and every ball is removed. >>>> Consider this rephrasing of the question: >>>> >>>> you have a set of n balls labelled 0...n-1. >>>> >>>> ball #m is added to the vase at time 1/2^(m/10) minutes >>>> before noon. >>>> >>>> ball #m is removed from the vase at time 1/2^m minutes >>>> before noon. >>>> >>>> how many balls are in the vase at noon? >>>> >>>> What does your "mathematics" say the answer to this >>>> question is, in the "limit" as n approaches infinity? >>> My mathematics says that it is an ill-posed question. And it doesn't >>> give an answer to ill-posed questions. >>> >>> Han de Bruijn >>> >> Actually, that question is not ill-posed, and has a clear answer. The >> vase will be empty, if there is any limit on the number of balls, and >> balls can be removed before more balls are added, but it is not the >> original problem, which states clearly that ten balls are inserted, >> before each one that is removed. That's the salient property of the >> gedanken. Any other scheme, such as labeling the balls and applying >> transfinitology, violates this basic sequential property, and so is a ruse. > > There are two equivalent truths: > > (1) (Each) ball number n will come out before noon. > (2) When ball number n comes out, more than n balls remain in the vase. > > Both are absolutely correct. This shows that one can not consistently > calculate with infinity. Yes, you can't calculate with something that isn't a number with measure of some sort. But, you can define infinite measures, such as a standard number of reals per unit interval. Then you can calculate. > > I published a similar but less lucid example as the 100 Euro question > on my homepage: > http://www.fh-augsburg.de/~mueckenh/Infinity/PreisfrageE%27.pdf > > For all sequences {2,4,6,...,2n} there exists z eps {2,4,6,...,2n} such > that > |{2,4,6,...,2n}| < z. But in the limit there is not such a z eps > {2,4,6,...}. > We have always |{2,4,6,...}| > z. Yes, because there is no real limit z defined. Over any given range of iterations or value, one can detect the difference in size of the set. Where it's defined as "finite", it has no clear boundary, and we get into this mess. > > Or briefly lim [n-->oo] 2n/n < 1. The first proof of this is worth 100 > Euro. > > Regards, WM > Do I get to divide by 0? ;) Just kidding. Tony
From: Virgil on 7 Oct 2006 23:53 In article <45287184(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David R Tribble wrote: > > Tony Orlow wrote: > >>> For the sake of this argument, we can talk about infinite reals, of > >>> which infinite whole numbers are a subset. > > > > David R Tribble wrote: > >>> Every member of N has a finite successor. Can you prove that your > >>> "infinite naturals" are members of N? > > > > Tony Orlow wrote: > >> Yes, if "finite successor" is the only criterion. > >> > >> To prove finiteness of such a string: > >> > >> The bits over each sequence are indexed by natural numbers, which are > >> all finite, yes? > >> > >> For any finite bit position, the string up to and including that bit > >> position can only represent a finite value, yes? > >> > >> Therefore, there is no bit position where the string can have > >> represented anything but a finite value, see? If the length is > >> potentially, but not actually, infinite, so with the value. > > > > So you're saying that finite bitstrings can only represent finite > > naturals. > > > > Strings with only finite bit positions. Wrong!!! Strings with only finite bit positions can still have infinitely many bit positions as there are infinitely many finite naturals. Finite naturals always have a finite most significant bit position and only finitely many non-zero digits. > > So obviously this rule, given a starting point of 0, a finite natural > > and a finite-length bitstring, can never produce anything but another > > finite-length bitstring as a successor. So you've proven that N > > can contain only finite naturals. > > > > Unless you think that your rule allows an infinite bitstring successor > > to be formed from some finite bitstring? > > You will not produce 1 bits in infinite positions without an infinite > number of successions. Which the naturals forbid. > > > > >> You don't really question why the successor to ...11110000 is equal to > >> ...11110001, do you? > > > > Again, I can't answer that until you define those numbers in a > > meaningful way. As you proved above, they are obviously not > > members of N. > > > > 1) ....00000 is a number. It is a digit string, and might be a numeral, but without a suitable context it is not a number. > 2) If x is a number, then the successive number, formed by inverting the > rightmost 0 and all 1's to the right of it, is also a number. And if it is NaN then neither are any of those other things.
From: Virgil on 7 Oct 2006 23:55 In article <45287451$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > Tony Orlow schrieb: > > > >> mueckenh(a)rz.fh-augsburg.de wrote: > >>> Tony Orlow schrieb: > >>> > >>> > >>>>>> Why not? Each and every number of the list terminates. That one is a > >>>>>> number > >>>>>> that does *not* terminate. > >>>>>> > >>>>>> > If you think that 0.111... is a number, but not in the list, > >>>>> It is me who insists that it is not a representation of a number. > >>>> Well, Wolfgang, that sets us apart, though I agree it's not a "specific" > >>>> number. It's still some kind of quantitative expression, even if it's > >>>> unbounded. Would you agree that ...333>...111, given a digital number > >>>> system where 3>1? > >>> That is the similar to 0.333... > 0.111.... But all these > >>> representations exist only potentially, in my opinion. The difference > >>> is, that 0.333... can be shown to lie between two existing numbers, so > >>> we can calculate with it, while for ...333 this cannot be shown. > >> I think it can be shown to lie between ...111 and ...555, given that > >> each digit is greater than the corresponding digit in the first, and > >> less than the corresponding digit in the second. > > > > Yes, but only if we define, for instance, > > > > A n eps |N : 111...1 < 333...3 where n digits are symbolized in both > > cases. > > > > This approach would be comparable with the "measure" which gives > > > > A n eps |N : |{1,2,3,...,2n}| = 2*|{2,4,6,...,2n}|. > > > > I don't know whether these definitions are of any use, but I am sure > > that they are not less useful than Cantor's cardinality. > > > > Regards, WM > > > > . > > > > My opinion about that is, if one wants to talk about what happens "at > infinity", that's the way that makes sense, not the measureless way of > abstract set theory. I trust limit concepts, but not limit ordinals. And those who trust both know better than to trust TO.
From: mueckenh on 8 Oct 2006 08:08
MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Of course. The question was: Did Cantor use ordinal subtraction. And > > can one define it under certain circumstances. > > > > k + omega = {-k, -k+1, ..., 0, 1 , 2, 3, ...} is different from > > > > -k + omega = {k, k+1, k+2, ...} > > > > I used it as an abbreviation to explain that the set omega (or the set > > of digit indexes of a certain number: 0.111...) is not unuiquely > > defined. > > As I said, I haven't read some of the earlier posts. > > So, I am still not familiar with your notation. > > Also, we need to be clear that, usually, by 'set theory' we mean one of > the axiomatized versions that came well after Cantor, such as Z, ZF, > ZFC, NBG, and other variants. > > So to see if your claims have any import for set theory as currently > practiced, it would help if you would give your definitions (in context > of the language of, say, Z set theory) and argument together in a > single post to which we could refer. > > Definitions then would be for '-' (as applied to ordinals), 'digit > indexes', and decimal notation (e.g., your '0.111...'). This example is but another variant of the vase, although not so clear. As you even are reluctant to accept the vase as a contradiction, it is useless to make any effort to translate this one. Take just: 0) There is a bijection between the set of balls entering the vase and |N. 1) There is a bijection between the set of escaped balls and |N. 2) There is a bijection between (the cardinal numbers of the sets of balls remaining in the vase after an escape)/9 and |N. (Instead of "balls", use "elements of X where X is a variable".) Regards, WM |