> 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.)
Talking about the infinite in that way is totally empty. The differences are in what one can do with that infinite. For example, using variables ranging ove the set of natural numbers. Or show the existence of a natural number with certain properties by contradiction - this requires much more "actual existence" of the set of naturals than merely using variables. In mathematics we at least implicitly know this. You don't, because you are too stupid for mathematics.