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Virgil wrote:
> 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.
Nor, obviously, do I. My point was that in your haste to try to beat
Tony on quantity, what you actually said was simply wrong. What you
said was that "[cardinality] is the only definition of set size that is
known to produce a valid partial ordering on sets." You do not, I
presume, claim this is actually true?
AAMOF, of course cardinality isn't the only definition of set size that
produces a total ordering on sets:
Let the set {} be "empty"
Let the set {x} for any x be "a singleton"
Let the set {a, b} for any a,b be "small"
Let the set {p, q, r } for any p q r be "medium"
And let all other sets be "large"
These sizes are totally ordered, are they not?
Brian Chandler
http://imaginatorium.org
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