Re: Non-aleph cardinals in set theory without axiom of choice?

On 2007-12-23, in sci.math, mike3 wrote:
But this doesn't seem to work. Wouldn't this imply the continuum
hypothesis is false, as then beth_1, the cardinal number of the
continuum, could not equal aleph_1, since it's not an aleph!
But that is independent of the Zermelo-Fraenkel axioms!

Right, and so is the negation of choice. In absence of choice the two usual
formulations of the continuum hypothesis:

"The cardinality of the continuum is aleph-1."

"Every infinite set of reals is either countable or of the cardinality of
the continuum."

are not equivalent. Even on the assumption that the beth-1 is not an aleph
-- which, incidentally, is not implied by failure of choice -- the second
formulation remains independent.

This would suggest at best we could say that whether or not beth_1 is
an aleph in ZF (note: no AC) is undecidable.

Right.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.

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