On Tuesday, June 11, 2013 4:09:56 PM UTC-7, Ralf Bader wrote: > firstname.lastname@example.org 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.
I doubt, WM can prove his A then B then A then ... algorithm satisfies that requirement. I will working proving it can't.