On 17 Jun., 22:14, MoeBlee <jazzm...@hotmail.com> wrote: > On Jun 17, 2:00 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> > That means, even undefinable real numbers are in trichotomy with each > > other and in particular with definable real numbers? > > You said trichotomy is not important in this discussion.
Oh, I cannot remember and I am surprised. Trichotomy is the most important issue when dealing with numbers. It is always important. > > Anyway, if 'definable real number' is given a definition in ZFC and > every 'definable real number' is a real number, then of course, > trichotomy holds. > > Listen, you don't need to waste our time. > > If I say trichotomy holds among all real numbers than I mean just what > I said.
But if there are more than countably many real numbers, then there are most of them so called moonlightr numbers. We do not know anything about them. How can they obey trichotomy?
So when you say what you said, then you should argue how you can do what you said. > > > How can that be? There is no way to name an undefinable real (because > > there are only countably many names). And it is impossible to define a > > real number by an infinite sequence, because onl finite sets and > > sequences can be defined by listing the elements or terms. > > We prove a theorem that the ordering satisfies trichotomy. > > All the rest of your perceived difficulties with this have no bearing > on that. Meanwhile, if you think we prove two contradictory theorems, > then just state the exact P such that you think P and ~P are proven in > ZFC.
You would not agree that proving trichotomy for numbers that have no definition and cannot be computed and cannot be constructed, is a contradiction? > > > Have you ever tried to put an undefinable real number in order with > > other, definable real numbers? > > Whatever I have or have not tried has no bearing on what is or is not > a theorem of ZFC.
But you could find out whether the theorem is sensible or is nonsense, if you only tried to apply it. > > > > Or to be pedantic: > > > > <x y> e <_r > > > <y x> e <_r > > > x=y > > > > And there is no ordering on the cardinals, > > > OK. Let us stay with the reals. Say you have two undefinable reals > > between 0 an 1. How can you manage to find out which one is less than > > the other? > > To say that trichotomy is a theorem is not to say also that we have a > way to find out anything at all. We have found out that there is a > proof of the trichotomy of the reals. That's all I claim in this > immediate regard.
But don't you think that it would be better to prove trichotomy by putting two numbers in correct order by magnitude than to prove that this can be done whereas in fact it cannot be done?