In article <firstname.lastname@example.org>, email@example.com wrote:
> On Saturday, 15 June 2013 11:01:16 UTC+2, Zeit Geist wrote: > > Virgil supplied a bijection between Q and N.
Actually, I have only supplied injections each way, but, at least outside of Wolkenmuekenheim, that is well known to be sufficient to establish existence of bijections. > > He ws not the first. But a eqinumerousity requires completeness. And that is > lacking
What sort of "completeness" is lacking?
The inclusion f(x) = x injects every n in |N to a different q in |Q.
The map g:|Q -> |N defined by g(0) = 1, and for m, n realatively prime naturals g(m/n) = 2^m*3^n and g(-m/n) = 5^m*7^n injects |Q into |N.
Thus there exist bijections between |N and |Q, at least everywhere outside of the wild weird world of WMytheology
> > > > > Also, I'm sure, whatever proof this is from, it's about Cardinals and > > > > not Ordinals. Do you know the difference? > > > Answer? Do you know > > > > the difference? > > That does not play a role in a case where we only > > > > have to show that every enumerated rational eventually will be taken > > > > into the well-ordered set. > Since you used "well-ordered" in that > > > > statement it does make a difference. > > You seem to think that there are magic forces? I *never* leave the domain of > finite sets (like Cantor never leaves it when enumerating something with > finite naturals).
Then WM's set of rationals must be a finite set, in contrast to everyone else's sets of rational, and FINITE ORDERED SETS ARE ALL WELL-ORDERED.
So in WM's wild weird world of WMytheology, ALL ordered sets are well-ordered. And are reverse well-ordered as well (every non-empty such set has a smallest member as well as a largest member).
That certainly must make WM's topology of the real number line a bit different, and removes any possibility of having infinite sequences or standard sort of continuity
> In the finite domain, there is no difference.
But in WM's finite domain there is a great deal of mathematics that simply does not take place.
Brouwer could, with difficulty and with at last the theoretical existence of actual infiniteness, make it work.
WM, with neither an actual infinity nor any such mathematical talent as Brouwer had, cannot come even close to making it work, so his WMytheology is a mathematical disaster area. --