query: Is Banach–Tarski paradox false without axiom of choice?
positive_passages.docid: 156161
positive_passages.text: Banach-Tarski theorem without axiom of choice
negative_passages.docid: 5035
negative_passages.text: Can unprovability unprovable? Is there an $\omega$-fold unprovability?
negative_passages.docid: 151215
negative_passages.text: Paradox of General Comprehension in Set Theory, other than Russell's Paradox
negative_passages.docid: 96717
negative_passages.text: Hahn-Banach Theorem for separable spaces without Zorn's Lemma
negative_passages.docid: 78290
negative_passages.text: Countable axiom of choice: why you can't prove it from just ZF
negative_passages.docid: 67259
negative_passages.text: Can we prove that a bounded closed subset of $\mathbb R^n$ is compact without Axiom of Choice?
negative_passages.docid: 78228
negative_passages.text: Hausdorff Maximal Principle and 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: 92614
negative_passages.text: Does the Banach-Tarski paradox contradict our understanding of nature?
negative_passages.docid: 88099
negative_passages.text: Not $\sigma$-compact set without axiom of choice
negative_passages.docid: 34674
negative_passages.text: Does the proof of Bolzano-Weierstrass theorem require axiom of choice?
negative_passages.docid: 87655
negative_passages.text: Consequence of the Hahn-Banach Theorem
negative_passages.docid: 77546
negative_passages.text: Does a "backwards" choice function imply the Axiom of Choice?
negative_passages.docid: 7214
negative_passages.text: Does linear ordering need the Axiom of Choice?
negative_passages.docid: 66573
negative_passages.text: Intuition behind why Stein's paradox only applies in dimensions $\ge 3$
negative_passages.docid: 85769
negative_passages.text: Do you need the Axiom of Choice to accept Cantor's Diagonal Proof?
negative_passages.docid: 147908
negative_passages.text: Why use the biconditional in the Axiom of Extensionality