query: The existence of uncountable regular cardinals in $ZF$
positive_passages.docid: 150605
positive_passages.text: Existence of a regular uncountable $\aleph_{\alpha}$ without $\mathsf{AC}$
negative_passages.docid: 113668
negative_passages.text: Uncountable chain of subsets of $\mathbb N$
negative_passages.docid: 140799
negative_passages.text: Prove that there exist uncountable family of subsets of $\mathbb{N}$ which is linearly (totally) ordered by relation of inclusion.
negative_passages.docid: 13924
negative_passages.text: For all infinite cardinals $\kappa, \ (\kappa \times \kappa, <_{cw}) \cong (\kappa, \in).$
negative_passages.docid: 66301
negative_passages.text: The cardinality of the set of countably infinite subsets of an infinite set
negative_passages.docid: 70196
negative_passages.text: Is there any well-ordered uncountable set of real numbers under the original ordering?
negative_passages.docid: 20085
negative_passages.text: What's the cardinality of all sequences with coefficients in an infinite set?
negative_passages.docid: 99599
negative_passages.text: Uncountable and countable
negative_passages.docid: 82947
negative_passages.text: Undecidable cardinality
negative_passages.docid: 9651
negative_passages.text: Cardinality of cartesian product of an infinite set with N
negative_passages.docid: 25685
negative_passages.text: What is the product of all nonzero, finite cardinals?
negative_passages.docid: 19924
negative_passages.text: Uncountable ordinals without power set axiom
negative_passages.docid: 15438
negative_passages.text: do all uncountable sets have same cardinality as real numbers?
negative_passages.docid: 63276
negative_passages.text: Is an arbitrary union of a chain of countable sets countable?
negative_passages.docid: 140868
negative_passages.text: Axiomatization of $\mathbb{Z}$ via well-ordering of positives.
negative_passages.docid: 154474
negative_passages.text: Image of countable and uncountable sets
negative_passages.docid: 83185
negative_passages.text: How to show countability of $\omega^\omega$ or $\epsilon_0$ in ZF?