query: List of theorems equivalent to choice
positive_passages.docid: 157490
positive_passages.text: Statements equivalent to the Axiom of Choice
negative_passages.docid: 17758
negative_passages.text: Theorem numbering
negative_passages.docid: 120252
negative_passages.text: Equivalences of bounded sets definitions
negative_passages.docid: 45878
negative_passages.text: Elementary Set Theory: Ordinals, Well Orderings and Isomorphisms
negative_passages.docid: 24478
negative_passages.text: What is the name of this theorem, and are there any caveats?
negative_passages.docid: 50235
negative_passages.text: Numbering theorem-like environments
negative_passages.docid: 73526
negative_passages.text: The equivalence of "Every surjection has a right inverse" and the Axiom of Choice
negative_passages.docid: 87955
negative_passages.text: How to prove that $P \rightarrow Q$ is equivalent with $\neg P \lor Q $?
negative_passages.docid: 81509
negative_passages.text: The cardinality of a countable union of countable sets, without the axiom of choice
negative_passages.docid: 17624
negative_passages.text: Axiom of Choice iff Every set has a choice function
negative_passages.docid: 141299
negative_passages.text: Alternative set theories
negative_passages.docid: 141814
negative_passages.text: Proofs using theorems instead of axioms
negative_passages.docid: 7119
negative_passages.text: box around theorem statement
negative_passages.docid: 34045
negative_passages.text: 3 Statements of axiom of choice are equivalent
negative_passages.docid: 65910
negative_passages.text: Reflexivity of equivalence relations
negative_passages.docid: 105303
negative_passages.text: 'Obvious' theorems that are actually false
negative_passages.docid: 130998
negative_passages.text: Is $\exists x \in A ~:~ P(x)$ the same as $\exists x ~:~ x \in A \implies P(x)$