Does a total or well ordering has a countable cofinal

Asked May 13 '23 at 17:58

Active May 13 '23 at 18:01

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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?

edited May 13 '23 at 18:01

asked May 13 '23 at 17:58

wsz_fantasy

1,672

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? – wsz_fantasy 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

Does a total or well ordering has a countable cofinal

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