[FOM] Another question about ZF without Choice

[Andres Caicedo]

Hi,

In a message dated Jan. 28 I asked whether Sierpinski's ZF result that
  aleph(X) < aleph(P(P(P(X)))) for all X
could be improved by replacing the triple power set with a double power 
set.

In a follow up dated Feb. 2 I indicated that one can, provided that
  aleph(X) is not aleph_alpha for some infinite limit ordinal
  alpha < aleph_alpha.

I recently found a reference that settles the other case, and wanted to 
give an update for those curious about the question. In Theorem 11 of John 
L. Hichman, "Lambda-minimal lattices", Zeitschr. f. math. Logik und 
Grundlagen d. Math., 26 (1980), 181--191, it is shown that for any such 
alpha it is consistent to have an X with
  aleph(X) = aleph_alpha = aleph(P(P(X))).

  Best,
  Andres