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I read a post sometime earlier asking about the cardinality of total ordering on a set, which I forgot about the detail but led me to this question.

For a total/well-ordered set $A$ does there exist a cofinal such that it is countable? If not, what is a counterexample?

My intuition is to use Zorn's lemma, which leads to the problem that the union of a chain of countable sets is not necessarily countable, but I do not completely understand the answer. Anyway, if we assume AC or perhaps even GCH, can we get any conclusion about the cardinality of the cofinal?

patchouli
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  • Depends on the ordering… sometimes yes and sometimes no. – spaceisdarkgreen May 13 '23 at 18:00
  • @spaceisdarkgreen Yeah sorry I fixed my question, what do you think about providing a counterexample then? – patchouli May 13 '23 at 18:02
  • The first uncountable ordinal. The size of the smallest cofinal subset is called the cofinality and the cofinality of an ordinal/well-ordered set can be any regular cardinal. – spaceisdarkgreen May 13 '23 at 18:04
  • So the example is $\omega_1$ ... see https://math.stackexchange.com/questions/956779/an-easy-to-understand-definition-of-omega-1 – GEdgar May 13 '23 at 18:17

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