Re: Set Theory question

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.

--
Stephen J. Herschkorn sjherschko@xxxxxxxxxxxx
Math Tutor on the Internet and in Central New Jersey and Manhattan

.

Relevant Pages

  • Re: Set Theory question
    ... if choice fails it's possible that every ... cardinal has cofinality omega." ... well-orderings of N is a well-ordered set of ordinality omega-1. ... countable ordinals was totally ordered. ...
    (sci.math)
  • Re: Set Theory question
    ... if choice fails it's possible that every ... cardinal has cofinality omega." ... well-orderings of N is a well-ordered set of ordinality omega-1. ...
    (sci.math)
  • Re: Set Theory question
    ... if choice fails it's possible that every ... cardinal has cofinality omega." ... number of countable ordinals is at most omega_1. ... Herman Rubin, Department of Statistics, Purdue University ...
    (sci.math)
  • Re: Set Theory question
    ... First, one poster (Aatu Koskensilta) ... if choice fails it's possible that every ... cardinal has cofinality omega." ... it's not true that all ordinals could have cofinality omega. ...
    (sci.math)