H. Herrlich and K. Keremedis wrote in their article "Remarks on the space $\aleph_1$ in ZF" (Topol. Appl. 158 (2011)) that CC($\mathbb{R}$) implies that the first uncountable cardinal number is regular. I wonder whether it can happen in a model for ZF that the first uncountable ordinal number is regular and, simultaneously, there is a non-void countable collection of non-void subsets of $\mathbb{R}$ which does not have a choice function. Perhaps, there are mathematicians who now a satisfactory answer to my question.
Try Cohen's basic model M1 in the consequences of the axiom of choice book by P. Howard and J. Rubin.
Dear Professor Keremedis,
I thank you for your answer very much. It is helpful. I shall look inside the book by Howard and Rubin.
Regards, Eliza
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