[FOM] Another question about ZF without Choice
caicedo at diamond.boisestate.edu
Tue Apr 28 11:49:42 EDT 2009
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
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))).
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