[FOM] Another question about ZF without choice
caicedo at diamond.boisestate.edu
Wed Jan 28 16:06:43 EST 2009
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
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.
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