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The Ghost In The Machine wrote:
> In sci.logic, Rupert
> <rupertmccallum@xxxxxxxxx>
> wrote
> on 25 Aug 2006 23:59:26 -0700
> <1156575566.502150.268470@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>:
> >
> > The Ghost In The Machine wrote:
> >> 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.
> >>
> >
> > Why not? However you define them, surely the subset relation would be a
> > partial ordering? Remember a partial ordering doesn't have to be
> > connected. Even the diagonal relation is a partial ordering.
>
> Hm...good point, if a bit on the thin side. :-)
>
> >
> >> 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.
> >>
> >
> > If R can be totally ordered, then of course every subset of a set
> > equipollent to R can be as well.
>
> Is R^3 equipollent to R?
>
Yes. R is equipollent to the set of infinite binary sequences B. B is
easily seen to be equipollent with BXB (mesh the two sequences
together). Hence R is equipollent to R^n for any positive integer n.
> >
> >> --
> >> #191, ewill3@xxxxxxxxxxxxx
> >> Windows Vista. Because it's time to refresh your hardware. Trust us.
> >
>
>
> --
> #191, ewill3@xxxxxxxxxxxxx
> Windows Vista. Because it's time to refresh your hardware. Trust us.
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