[FOM] Another question about ZF without choice

[Thomas Forster]

Aki Kanamori has a fairly recent article on this which i think has
the most recent information.    I think he is on this list and may
reply.  There is also Halbeisen-Shelah.   I'm afraid i haven't got
exact citations to hand but they shouldn't be hard to find.

    tf

On 
Wed, 28 Jan 2009, Andres Caicedo wrote:

>
>  Hi,
>
>  I seem to recall that the following is known, but I have no idea why I
> would have come across it in the past. I haven't been able to
> find a reference or produce an example myself, so I would appreciate any
> pointers, hints, and/or historical remarks.
>
>  Recall that the aleph of a set X, aleph(X), is the smallest ordinal
> (necessarily, a cardinal) that does not inject into X.
>
>  One can check that aleph(X) injects into P(P(P(X))), the triple powerset
> of X.
>
>  I would like an example where aleph(X) does not inject into P(P(X)).
>
>  This seems to be slightly subtle; for example, there is such an injection
> if X is Dedekind-finite, or if X is equipotent with a square.
>
>  (But choice is equivalent to every infinite cardinal being a square).
>
>  I confess I haven't thought about this for a decent amount of time, and I
> apologize if the question is trivial. But I am curious, and would very
> much like to know.
>
>  Thanks!
>
>  Best,
>  Andres
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