query: Axiom of choice and Euclid's first proposition
positive_passages.docid: 27025
positive_passages.text: How do I choose an element from a non-empty set?
negative_passages.docid: 78148
negative_passages.text: Why does the infinite prisoners and hats puzzle require the axiom of choice?
negative_passages.docid: 3663
negative_passages.text: Finding counterexamples in elementary set theory.
negative_passages.docid: 4383
negative_passages.text: Finitely generated monoids - how to show that there is not first order set axiomatizing them.
negative_passages.docid: 87451
negative_passages.text: Construction of matrices under ZFC axioms
negative_passages.docid: 7214
negative_passages.text: Does linear ordering need the Axiom of Choice?
negative_passages.docid: 96504
negative_passages.text: First-Order Logic vs. Second-Order Logic
negative_passages.docid: 132046
negative_passages.text: Looking for Cantor's original proof of the Cantor-Bernstein theorem that relies on the axiom of choice?
negative_passages.docid: 142840
negative_passages.text: Open axioms of equality
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: 139678
negative_passages.text: Advantage of accepting the axiom of choice
negative_passages.docid: 109325
negative_passages.text: Euclid Lemma proof reasoning
negative_passages.docid: 118189
negative_passages.text: Proving the pairing axiom from the rest of ZF
negative_passages.docid: 126870
negative_passages.text: Show that axiom of replacement implies from the Axiom of Specification [Proof Verification]
negative_passages.docid: 50506
negative_passages.text: What concept does an open set axiomatise?
negative_passages.docid: 29755
negative_passages.text: Power set equivalency of Axiom of Choice
negative_passages.docid: 115675
negative_passages.text: Strange consequences of Axiom of Choice in Zermelo set theory