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In article <1162276209.968986.110750@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
imaginatorium@xxxxxxxxxxxxx wrote:
> Virgil wrote:
> > In article <1162268163.368326.64650@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
> > imaginatorium@xxxxxxxxxxxxx wrote:
> >
> > > Virgil wrote:
> > > > In article <45462ba0@xxxxxxxxxxxxxxxxxxx>,
> > > > Tony Orlow <tony@xxxxxxxxxxxxx> wrote:
> > > >
> > > > > stephen@xxxxxxxxxx wrote:
> > > > > > Tony Orlow <tony@xxxxxxxxxxxxx> wrote:
> > > > > >> stephen@xxxxxxxxxx wrote:
> > > > > >>> Tony Orlow <tony@xxxxxxxxxxxxx> wrote:
> > > > > >>>> stephen@xxxxxxxxxx wrote:
> > >
> > > <snipola>
> > >
> > > > > >> OH. So, sets don't have sizes which are numbers, at least at
> > > > > >> particular
> > > > > >> moments. I see....
> > > > > >
> > > > > > If that is what you meant, then you should have said that.
> > > > > > And technically speaking, sets do not have sizes which are numbers,
> > > > > > unless by "size" you mean cardinality, and by "number" you include
> > > > > > transfinite cardinals.
> > > > >
> > > > > So, cardinality is the only definition of set size which you will
> > > > > consider.....your loss.
> > > >
> > > > It is the only definition of set size that is known to produce a valid
> > > > partial ordering on sets.
> > >
> > > Huh? I thought cardinality produced a valid *total* ordering on sets.
> >
> > The cardinalities are totally ordered, but the sets are not.
> > A total order on sets would require that when neither of two sets was
> > "larger than" the other that they must be the same set, not merely the
> > same size.
>
> Oh, right. But - and I'm not quite sure how to say this, but the
> cardinalities _are_ totally ordered; for any two sets A and B, c(A) <
> c(B), or c(A) = c(B), or c(A) > c(B). If you "reduce" the sets by the
> cardinality equivalence relation, they are totally ordered. The subset
> relation doesn't lead to an equivalence relation, only a partial
> ordering: so there is no s(A) = s(B) unless A=B; but for most pairs of
> sets, the subset relation simply says nothing at all. (Until His
> Master's Voice is heard, telling us something totally arbitrary.)
>
> Anyway, your claim was clearly wrong, since the subset relation
> provides a valid partial ordering on sets.
Since there are sets that the subset relation cannot compare for "size",
since trichotomy does not hold for the subset relation, I do not regard
the subset relation as a valid measure of size in the same sense that
cardinality is, at least assuming the axiom of choice.
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