Re: abundance of irrationals!)


Martin Shobe wrote:

> >Counting a set of finite cardinals {1,2,3,...,n} is the same as
> >counting a set of finite ordinals {1,2,3,...,n}, because finite
> >cardinals and finite ordinals are the same. The result is obviously
a
> >finite cardinal number.
>
> Not Ok.
>
> The corrected statement would be
> "Counting a [finite] set of finite cardinals {1,2,3,...,n} is the
same
> as counting a [finite] set of finite ordinals {1,2,3,...,n},

Why do you require finite set? There is absolutely no justification by
your following argument, which concerns ONLY the finity of the numbers
themselves, not how many there are.

> because
> finite cardinals and finite ordinals [have an obvious bijection
> between them.] The result is obviously a finite cardinal number.

Do you see that your "finite set" inserted above does not appear in the
reasoning?

> Things change when you get to the infinite.
> Even one as small as aleph_0.

Of course. But why then do you believe that the bijection
1 - 2
2 - 4
3 - 6
....
does not change *in the infinite*, and that both cardinalities are
equal? Because all these numbers are finite numbers, I assume. But the
ordinal numbers in my bijection are all finite too.

Any idea why there should be a difference?

Regards, WM

.

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