query: From one version of Axiom of Choice to another
positive_passages.docid: 17624
positive_passages.text: Axiom of Choice iff Every set has a choice function
negative_passages.docid: 35123
negative_passages.text: How do the separation axioms follow from the replacement axioms?
negative_passages.docid: 27600
negative_passages.text: Gödel's (in)completeness theorems and the axiomatization of Euclidean geometry
negative_passages.docid: 38242
negative_passages.text: Explicit bijection between the successor of $\alpha$ and $\alpha$ (infinite ordinal).
negative_passages.docid: 53025
negative_passages.text: Why the need of Axiom of Countable Choice?
negative_passages.docid: 139678
negative_passages.text: Advantage of accepting the axiom of choice
negative_passages.docid: 121847
negative_passages.text: Don't we need the axiom of choice to choose from a non-empty set?
negative_passages.docid: 9158
negative_passages.text: Proving that a propositional theory of any cardinality has an independent set of axioms
negative_passages.docid: 153935
negative_passages.text: A weaker Axiom of Infinity?
negative_passages.docid: 73526
negative_passages.text: The equivalence of "Every surjection has a right inverse" and the Axiom of Choice
negative_passages.docid: 49220
negative_passages.text: Open Sets of $\mathbb{R}^1$ and axiom of choice
negative_passages.docid: 144899
negative_passages.text: Do We Need the Axiom of Choice for Finite Sets?
negative_passages.docid: 155207
negative_passages.text: Confusion regarding one formulation of the Axiom of Choice.
negative_passages.docid: 104771
negative_passages.text: Dedekind's cut and axioms
negative_passages.docid: 78290
negative_passages.text: Countable axiom of choice: why you can't prove it from just ZF
negative_passages.docid: 81509
negative_passages.text: The cardinality of a countable union of countable sets, without the axiom of choice
negative_passages.docid: 153634
negative_passages.text: A statment about the equivalence between the Axiom of Choice and the surjectivity of the projections.