Re: Orlow cardinality question

stephen@xxxxxxxxxx said:
> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
> > stephen@xxxxxxxxxx said:
> >> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
> >> > Randy Poe said:
> >> >>
> >> >>
> >> >> Tony Orlow (aeo6) wrote:
> >> >> > Randy Poe said:
> >> >> > > Again, you are insisting on an axiom that doesn't exist,
> >> >> > > in this case that if A < B, then |A| < |B|. That is, if
> >> >> > > A is a proper subset of B, then the cardinality of A is
> >> >> > > smaller than the cardinality of B. As you have been told
> >> >> > > many times, there is no such requirement on infinite sets,
> >> >> > > and indeed a property of infinite sets is that there
> >> >> > > exists A < B with |A| = |B|.
> >> >> > Cardinality purports to describe the sizes of infinite sets.
> >> >>
> >> >> Bzzzzt. Wrong from the starting gate.
> >> >>
> >> >> Cardinality purports to be a method to assign an ordering
> >> >> to infinite sets. This it does remarkably well.
> >> > An ordering in terms of what? Element values? Standard deviation? No. It
> >> > purports to be a way of comparing the sizes of sets, that is, it purports to
> >> > distinguish between a few different infinite numbers of elements, and to prove
> >> > that no other distinction can be made, which is wrong.
> >>
> >> > Try defining the word cardinality in five words or less.
> >>
> >> That would be silly. It has a precise mathematical definition.
> >> Using anything other than the precise mathematical definition
> >> is pointless, unless you just want to make vague hand wavy
> >> arguments.
> >>
> >> Stephen
> >>
> > So, you can't summarize what it is that cardinality is supposed to accomplish,
> > but claim it is other than determining the size of a set? What is the basic
> > purpose of this theory, if not to measure sets?
> > --
> > Smiles,
>
> > Tony
>
> Cardinality is not a theory. It is a definition. Two
> sets have the same cardinality if there exists a bijection between them.
> I am not sure what you mean by "purpose", or what you would
> accept as a valid answer. What is the purpose of the "size of a set"?
>
> Stephen
>
Size is one measure of a set. The purpose of a set is to act as a collection.
That collection may have many attributes, but as a set, one attribute it always
has is its size, or the number of elements it includes. The problem that is
supposedly addressed by cardinality is the measure of set size, for both finite
and infinite sets, hopefully addressed in some consistent manner.
Unfortunately, it is not consistent enough for my tastes, with all the
exceptions claimed for infinite sets, as if they aren't even sets anymore, and
as if cardinality is suddenly NOT supposed to be a measure of set size. If
cardinality of infinite sets is not a measure of their size, if adding or
subtracting elements doesn't change the cardinality, then what the heck is it
supposed to signify? That's what I'm asking.
--
Smiles,

Tony
.

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