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?
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.
Have you ever tried to put an undefinable real number in order with other, definable real numbers? > > 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?