In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 9 Dez., 22:29, Virgil <vir...@ligriv.com> wrote: > > > > With respect to real numbers we need not only the answer to the > > > question whether x is a real number, but we need trichotomy with > > > respect to every real number. Undefinable numbers cannot satisfy this > > > need. > > > > WM may need it, but if the issue is only whether something is or is not > > a real number, there are definitions of what is a real number that does > > need not establish its trichotomy with anything. > > > > Trichotomy is a derived property not a definitional one of some > > definitions of the reals. > > Then you have another idea of real number. Be happy with it. For doing > mathematics including real calculations trichotomy is indispensable.
To say that a property is derived does not mean that it does not exist, as you ignorantly seem to be assuming, merely that it follows from other properties, rather than being a necessary part of the definition.
The trichotomy of the reals follows from other properties.
As far as I am aware, in both the Cauchy sequence definition of the reals and the Dedekind cut definition of the reals, trichotomy as a property of reals must be *deduced* from those definitions.
That it can be deduced and had been for both definitions does not justify WM's arguments. --