query: The Sequence Lemma and the Axiom of Countable Choice
positive_passages.docid: 150385
positive_passages.text: The use of the Axiom of Choice in this proof of the sequence lemma
negative_passages.docid: 19414
negative_passages.text: Uncountable subset with uncountable complement, without the Axiom of Choice
negative_passages.docid: 120064
negative_passages.text: Proving that without the axiom of choice there is a set with an accumulation point that isn'tthe limit of a convergeant sequence
negative_passages.docid: 5313
negative_passages.text: Existence of how many sets is asserted by the axiom of choice in this case?
negative_passages.docid: 127123
negative_passages.text: Cardinality of power set and binary sequence
negative_passages.docid: 73526
negative_passages.text: The equivalence of "Every surjection has a right inverse" and the Axiom of Choice
negative_passages.docid: 105815
negative_passages.text: Does proving (second countable) $\Rightarrow$ (Lindelöf) require the axiom of choice?
negative_passages.docid: 103769
negative_passages.text: The Cartesian product of a finite number of countable sets is countable
negative_passages.docid: 109926
negative_passages.text: Axiom of Regularity and infinite sequences
negative_passages.docid: 65308
negative_passages.text: Axiom of Choice: Family of non-empty sets mutually disjoint or not?
negative_passages.docid: 45878
negative_passages.text: Elementary Set Theory: Ordinals, Well Orderings and Isomorphisms
negative_passages.docid: 102202
negative_passages.text: The smallest infinity and the axiom of choice
negative_passages.docid: 68994
negative_passages.text: Countable closed sets
negative_passages.docid: 134844
negative_passages.text: Summablity and countability
negative_passages.docid: 38242
negative_passages.text: Explicit bijection between the successor of $\alpha$ and $\alpha$ (infinite ordinal).
negative_passages.docid: 153652
negative_passages.text: Countable ordinals are embeddable in the rationals $\Bbb Q$ -- proofs and their use of AC
negative_passages.docid: 94413
negative_passages.text: Proving Dedekind finite implies finite assuming countable choice