Re: Non-aleph cardinals in set theory without axiom of choice?

On Dec 22, 9:40 pm, mike3 <mike4...@xxxxxxxxx> wrote:
On Dec 22, 8:35 pm, Dave Seaman <dsea...@xxxxxxxxxxxx> wrote:

On Sat, 22 Dec 2007 18:16:30 -0800 (PST), mike3 wrote:
Hi.
I saw this:
http://en.wikipedia.org/wiki/Cardinal_number
It says that "if the axiom of choice fails there are additional
infinite cardinals that are not alephs".
So, could one provide an example of a set that has such a non-aleph
cardinality, in set
theory where the axiom of choice does fail? If so, what is it?

If AC fails, then there are sets that cannot be well ordered.  No such
set can have an aleph cardinality, because each aleph is an ordinal and
therefore is a well ordered set.

Ah. What would be an example of such a set then?

The set of all real numbers?
.

Relevant Pages