[FOM] Another question about ZF without Choice

todunion@ucollege.edu todunion at ucollege.edu
Fri Jan 30 19:44:04 EST 2009

For a simple (one-page) proof of the result sought below in ZF without
Choice see p. 550 of the article in The American Mathematical Monthly
(MAA),vol 109, No. 6, (June-July, 2002) by Leonard Gillman (pp544-553)

"Two Classical Surprises Concerning the Axiom
of Choice and the Continuum Hypothesis"

best regards,
Tom Dunion

On Wed, 28 Jan, 2009, Andres Caicedo wrote

>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.

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