On Jun 17, 2:00 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > On 17 Jun., 19:36, MoeBlee <jazzm...@hotmail.com> wrote: > > > > > Re: Jun 17, 12:23 pm, MoeBlee <jazzm...@hotmail.com>e: > > > P.S. > > > If the discussion about the formal theory ZFC, then I don't know what > > lack of trichotomy you're referring to. > > > Of course, many relations fail to satisfy trichotomy. But the standard > > strictly less than ordering on the reals satisfies trichotomy: > > > If x and y are reals, then exactly one of these holds: > > > x <_r y > > y <_r x > > x=y > > > where '<_r' stands for the standard stricly less than relation on the > > set of real numbers. > > 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.
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.
> 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.
> 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.
> > 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.
Look, please don't waste our time.
I said clearly that it is not at issue with me that you may find many contradictions between ZFC and your own mathematical notions. Such contradictions that you, in particular, find, are not of particular interest to me at this time.
On the other hand, you say you do not care about formal theories.
So there is little for us to discuss.
On the other hand, when you do talk about ZFC or about common arguments that are also formalized in ZFC, then I may wish to comment on your remarks. If you have a contradiction between ZFC and your mathematics, then you don't have to convince me since I well recognize that you may find such contradictions, but if you claim something "wrong" in ZFC itself, then please produce an actual contradiction IN ZFC or else we are just back to the fact that you find contradictions between ZFC and your own mathematics.