Re: Every set can be ... ordered?

The Ghost In The Machine wrote:
MoeBlee wrote:
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.

I hope I'm not mistaken that Rupert is correct in saying that I was
correct: The subset relation on a set is a partial ordering of the set.

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.

I hope I'm not mistaken that the standard ordering on the reals is a
total ordering of the reals.

MoeBlee

.

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