On 5 Feb., 16:53, fom <fomJ...@nyms.net> wrote: > On 2/5/2013 4:11 AM, WM wrote: > > > On 5 Feb., 01:26, Virgil <vir...@ligriv.com> wrote: > > >> If, as WM claims, there are at most countably many ways of accessing > >> reals and, as Cantor claimed, there are more than countably many reals, > >> then ... > > > Then the axiom that every real can be put in trichotomy with every > > rational number is contricted. Then ZFC contradicts analysis. > > In the logical type hierarchy that *defines* a real number, > the order relation of the rationals is inherited. The > rationals that are reals are distinct from the rationals > from which the real numbers are defined.
Please name a difference that can be proved to exist in mathematics. Same is untrue for naturals and positive finite cardinals. There is not the least difference with respect to mathematics.
> In this construction, > the identity of a real number is tied to the trichotomy of > the underlying rationals and to the fact that in the complete > space any given pair of irrationals taken to be distinct are > linearly separated by a rational.
That fact already proves that there cannot exist more irrationals than rationals. > > The problem of identity of a real number as part of the > real number system relative to identity within > Zermelo-Fraenkel set theory is a pseudo-metrization > problem.
No, it is the clever trick of selling matheological nonsense as ideas of intelligent heads to innocent believers in the existence of intelligent mathematicians.