[FOM] Another question about ZF without choice

[Andres Caicedo]

  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