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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.
Brian Chandler
http://imaginatorium.org
As I have pointed out when all this started, of course everyone knows
that the subset relation produces a perfectly valid *partial* ordering
on sets.
Brian Chandler
imaginatorium.org">http://imaginatorium.org
If that's what a partial ordering vs. a total ordering is, Bigulosity is
a partial ordering on sets, not total ordering. Different sets can have
the same Bigulosity.
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