Robert Coe wrote:
On Fri, 5 Apr 2013 22:17:01 +0200, Wolfgang Weisselberg
: Robert Coe wrote:
: On Tue, 2 Apr 2013 18:37:25 +0200, Wolfgang Weisselberg

: : I shouldn't bother, but I'm going to call your bluff. Please tell us what the
: : first two infinite numbers are

: [...]

: No, they are not sets. They are the (infinite) numbers of *members*
: incorporated in two different sets. I leapt to the conclusion that transfinite
: arithmetic is beyond your reach, and you seem to have proven me correct.

: So tell me, what are the first two infinite numbers? You
: seem to be bursting to tell everyone how smart you are!

No,

Ah, you're playing teacher.
So what are your credentials?

I'm giving you an opportunity to show us how well you understand
transfinite arithmetic. How can it be that one of the numbers (aleph-1) is
larger than the other (aleph-0), given that both of them are infinite?

countable, not countable.
Interval between *any* 2 real numbers always contains at least
one real number (a+b)/2, which by induction means they all
contain infinite real numbers.
Can't bijectively map from aleph-null to, say, interval [0-1[
in real numbers.

: BTW, in an English-speaking newsgroup, you really should refer to "aleph-one"
: rather than "aleph-eins". (Auf Deutsch, aleph-eins ist richtig,
: natürlich.)

: So you'd advocate "aleph-zero" in an English-speaking newsgroup,
: too?

Not necessarily (although I have heard it called that). The word "null" is
just as understandable in English as it is in German. In the math class where
I first encountered the number, the professor called it "aleph-null". And
aleph-1 was referred to as "c", which stood for the "cardinal number of the
continuum". I'm not even certain that at the time (the late 1950s) it had been
proven that there are no other numbers between aleph-0 and c. I've since read
that it's now known to be the case (which may account for the adoption of the
term "aleph-1"), but I don't recall seeing the proof.

Do a google on that proof and report back. That's your
opportunity to show us how well you remember transfinite
arithmetic.

-Wolfgang