Virgil
Posts:
4,479
Registered:
1/6/11
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Re: Mathematics in brief
Posted:
Dec 9, 2012 9:41 PM
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In article
<488be209-12d5-470c-b4d7-fcc23b4ce9e6@x3g2000yqo.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> 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.
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