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In sci.logic, MoeBlee
<jazzmobe@xxxxxxxxxxx>
wrote
on 25 Aug 2006 16:39:22 -0700
<1156549162.644162.249500@xxxxxxxxxxxxxxxxxxxxxxxxxxx>:
> MoeBlee wrote:
>> And what about "every set can be partially ordered"?
>
> Oops, nevermind that question. Obviously, every set is partially
> ordered by the subset relation on the set.
>
> MoeBlee
>
Every *power* set, maybe. But the reals wouldn't be able
to be ordered that way.
Of course it's easy enough to total order the reals;
if each real number r is defined by at least one Cauchy
sequence of rational numbers, then another real number
s can also be defined by another Cauchy sequence, and if
it is the case that there exists an M and a rational
d > 0 such that for every i,j > M, r_i > s_j + d, then r > s.
Or something like that.
Contrariwise, I am not sure if anyone's proven that one cannot
total-order a convex R^2 or R^3 subset.
--
#191, ewill3@xxxxxxxxxxxxx
Windows Vista. Because it's time to refresh your hardware. Trust us.
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