Re: Jun 17, 12:23 pm, MoeBlee <jazzm...@hotmail.com>e:
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.
Or to be pedantic:
<x y> e <_r <y x> e <_r x=y
And there is no ordering on the cardinals, since there is no set that has all cardinals in it, but still, we have the theorem (rendered here in English and symbols; as I'll not make that note usually now):
If X and Y are cardinals, then exactly one of these three:
X <_c Y Y <_c X X = Y
where '<c' stands for the cardinal strictly less-than predicate.