Re: Set Theory question

In article <45A156C9.7060903@xxxxxxxxxxxx>,
Stephen J. Herschkorn <sjherschko@xxxxxxxxxxxx> wrote:
Herman Rubin wrote:

In article <1168162902.018715.210800@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Jules <julianrosen@xxxxxxxxx> wrote:




I have a couple of questions. First, one poster (Aatu Koskensilta)
wrote "But interestingly, if choice fails it's possible that every
cardinal has cofinality omega." Did you mean that every ordinal has
cofinality omega? If not, then how is the cofinality of a cardinal
defined?



This is not so. Omega_1 does not have cofinality omega.


Careful, Herman. We are talking ZF (no choice) here. According to
Kunen (p. 30), Levy showed that it is consistent with ZF that omega-1
has cofinality omega.

You are right on this. However, the limit of a countable
number of countable ordinals is at most omega_1. To get
around this, one needs to have a weak form of the Axiom
of Choice.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.

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