Re: Orlow cardinality question

In article <MPG.1d20cbbdddf9357d989e4c@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:

> Virgil said:
>
> > One measure has been accurately applied. Anything that requires
> > knowledge of the properties of elements other than whether they are
> > members of the sets being compared is measuring something else
> > besides the sets themselves.

> And yet, you have no measure of cardinality for infinite sets without
> it. You cannot characterize an infinite set of numbers without
> reference to their quantities, or an infinite set of strings without
> reference to their characters or meanings.

Actually, I can characterise a set as being finite or infinite without
any reference at all to the type of members that it has, but only with
reference to whether it allows of any injection into any proper subset.

> >
> > What kind of elements a set contains is irrelevant to the "number"
> > of those elements for EVERY finite set. Why should that suddenly
> > change for infinite sets? TO has given no better reason than that
> > his guts tell him so. I put little trust in the mathematical
> > validity of TO's gut reactions.

> Because the way we define an infinite set is not by exhaustively
> enumerating all its elements, but by producing some kind of abstract
> system for generating them. This abstract generator is what defines
> the elements of the set and the set itself, and the measure of the
> set cannot be accomplished without refernce to this generator.

There are at least two quite different methods for "generating" the set
of reals, Dedekind cuts and equivalence classes of Cauchy sequences, in
which the objects generated are quite different, but one can still prove
a bijection between them.

So that the cardinality of the reals must be independent of the method
of generation, and not require any reference to it.
.

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