In article <email@example.com>, Zeit Geist <firstname.lastname@example.org> wrote:
> On Tuesday, June 11, 2013 4:09:56 PM UTC-7, Ralf Bader wrote: > > email@example.com wrote: > > > > > > > > > On Tuesday, 11 June 2013 20:43:30 UTC+2, Zeit Geist wrote: > > > > > > > > > > > I think your algorithm fails at the limit stage. > > > > > > > > > > That's the same with enumerating the rationals and with the diagonal of > > > > > the list. No limits available in all these cases. > > > > > If we enumerate the rationals, we do it up to n (since there is no > > > > > infinite natural number). > > > > > > > > No. Maybe you try to do it so, because you are stupid. WE might do something > > > > like this: We note that a positive rational r can be written as > > > > r = p_1^n1 * p_2^n2 * ... * p_k^nk, i.e. a product of powers of primes p_i > > > > with integer exponents ni. And then we map r to p_1^f(n1) * p_2^(n2) * ... > > > > * p_k^f(nk), where f(n) = 2n if n>0 and f(n) = -2n-1 if n<0 or so. We take > > > > some care with the cases r=0, r=1, ni=0 and then we can show that we have > > > > got a bijection from the positive rationals to the naturals. > > > > > > > > There is nothing WE do "up to n". There is no limit involved. What we need > > > > are variables ranging over various sets of numbers. I don't know whether a > > > > set of numbers must "actually exist" in order to be allowed to use > > > > variables ranging over that set. > > > > > > > > And I don't care about any opinions philosophers might hold about this > > > > point. For example > > > > http://www.leaderu.com/truth/3truth11.html > > > > is anything but convincing. There one reads: > > > > "2. The universe began to exist. > > > > > > > > 2.1 Argument based on the impossibility of an > > > > actual infinite. > > > > > > > > 2.11 An actual infinite cannot exist. > > > > 2.12 An infinite temporal regress of > > > > events is an actual infinite. > > > > 2.13 Therefore, an infinite temporal > > > > regress of events cannot exist. " > > > > Then some arguments for the alleged non-existence of an actual infinite are > > > > derived from Hilbert's hotel. But this is completely beyond the point, > > > > those arguments at best saying that the events in that infinite regress can > > > > not be shuffled around like the guests in that hotel. And indeed they > > > > can't, because they are in a fixed order and no events can b checked in or > > > > out of the past. (But even if the arguments given for the non-existence of > > > > actual infinities would apply, they still would not be convincing.) > > > > You mention of Hilbert's Hotel gives me an idea of how to show that the > rational numbers cannot be order isomorphic to the naturals. I don't have > all the details, but here's a sketch of it. > > A bus containing all the rationals in [0,1] pulls up to Hilbert's Hotel. > The Bellhop asks, "How did you all fit in that little bus?" > Someone replies, "Well, we can pack ourselves in rather densely." > ; ) > > The rationals know they can fit in the Hotel. After they are only > countably infinite. But they have a special request. They really > like to be nearby their "close" friends. So they ask: > > "Can you guarantee that if I give you a real number, d, then you can > give me a number number m, such that for any rational number, r, > If a rational x is in ( r - d, r + d ), > then x's room will be within m rooms of r's? > > The Bellhop looks at It and says > "Wow, you guys really are dense." > > If we want an order isomorphism between the rationals and the > naturals, the Topological property of "closeness" must be preserved.
The standard order for the naturals is a well-ordering.
The standard ordering for the rationals is a dense ordering.
No ordered set of more than one element can have both orderings, because any such a dense ordering then produces an infinite decreasing sequence which violates any well-ordering. > > I doubt, WM can prove his A then B then A then ... algorithm satisfies > that requirement. I will working proving it can't. > > > ZG --