Re: A CHALLANGE TO CANTORIANS
- From: The Ghost In The Machine <ewill@xxxxxxxxxxxxxxxxxxxxxxx>
- Date: Tue, 21 Aug 2007 20:56:20 -0700
In sci.math, tommy1729
<tommy1729@xxxxxxxxx>
wrote
on Mon, 20 Aug 2007 18:32:24 EDT
<27514796.1187649174311.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>:
william hughes wrote :
On Aug 20, 5:15 pm, tommy1729 <tommy1...@xxxxxxxxx>
wrote:
<snip>
professor ullrich ??????
he has NEVER HEARD OF 2^aleph_0 >= aleph_1
Piffle. He is well aware of the inequality.
However, you did not claim an inequality, you
claimed that the *equality*
2^aleph_0 = aleph_1
is true (if you doubt this look at your original
post).
Professor Ullrich is well aware that this equality
goes
by the name of the Continuum Hypothesis.
i doubt that...
On the other hand you seem unaware that
2^aleph_0 = c
so that when you said
2^aleph_0 = aleph_1
this is equivalent to saying
aleph_1 = c
lol
you made the assumption
aleph_1 = c
wich is not provable
I'm not sure if this equality provable or not, but you
win on a technicality.
The *beths* are the ones such that
beth_0 = card(N)
beth_1 = card(P(N)) = card(2^N)
beth_2 = card(P(P(N))) = card(2^(2^N))
etc.
The alephs are simply the transfinites in their total ordering,
yielding a sequence. That is to say, we know that
aleph_0 = card(N) < aleph_1 < aleph_2 < ...
AFAIK, the question of where beth_1 is in this sequence
has never been satisfactorially resolved.
Ditto for c = card(R).
[rest snipped]
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