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From: Leonid Lenov on 12 Jan 2010 07:32
Hello,
How much is Aleph_0 ^ Aleph_0? Is it equal to Aleph_1?
Thanks in advance.
From: Tonico on 12 Jan 2010 07:39
On Jan 12, 2:32 pm, Leonid Lenov <leonidle...(a)gmail.com> wrote:
> Hello,
> How much is Aleph_0 ^ Aleph_0? Is it equal to Aleph_1?
> Thanks in advance.


Put A:= Aleph_o, c = the continuum cardinality,then:

c = 2^A <= A^A <= (2^A)^A = 2^(A^2) = 2^A = c

By Bernstein-Schroeder's theorem, A^A = c

Tonio
From: William Elliot on 12 Jan 2010 07:47
On Tue, 12 Jan 2010, Tonico wrote:
> On Jan 12, 2:32�pm, Leonid Lenov <leonidle...(a)gmail.com> wrote:

>> How much is Aleph_0 ^ Aleph_0? Is it equal to Aleph_1?

Yes, assuming the continuum hypothesis.

> Put A:= Aleph_o, c = the continuum cardinality,then:
>
> c = 2^A <= A^A <= (2^A)^A = 2^(A^2) = 2^A = c
>
> By Bernstein-Schroeder's theorem, A^A = c

Theorem. GCH.
Proof. Occam's Razor.
From: Leonid Lenov on 12 Jan 2010 08:29
On Jan 12, 1:47 pm, William Elliot <ma...(a)rdrop.remove.com> wrote:
> On Tue, 12 Jan 2010, Tonico wrote:
> > On Jan 12, 2:32 pm, Leonid Lenov <leonidle...(a)gmail.com> wrote:
> >> How much is Aleph_0 ^ Aleph_0? Is it equal to Aleph_1?
>
> Yes, assuming the continuum hypothesis.

If we do not assume the continuum hypothesis is Tonico's argument
still valid to show that Aleph_0 ^ Aleph_0 <= Aleph_1?
From: A N Niel on 12 Jan 2010 09:33
In article
<6f00e4d9-ae47-4a4a-97d3-1d245d84e017(a)c3g2000yqd.googlegroups.com>,
Leonid Lenov <leonidlenov(a)gmail.com> wrote:

> On Jan 12, 1:47�pm, William Elliot <ma...(a)rdrop.remove.com> wrote:
> > On Tue, 12 Jan 2010, Tonico wrote:
> > > On Jan 12, 2:32�pm, Leonid Lenov <leonidle...(a)gmail.com> wrote:
> > >> How much is Aleph_0 ^ Aleph_0? Is it equal to Aleph_1?
> >
> > Yes, assuming the continuum hypothesis.
>
> If we do not assume the continuum hypothesis is Tonico's argument
> still valid to show that Aleph_0 ^ Aleph_0 <= Aleph_1?

No, in ZF it shows aleph_0 ^ aleph_0 = c >= aleph_1 .
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