query: Can a weak form of the union axiom be proved in ZF set theory?
positive_passages.docid: 27899
positive_passages.text: Can we prove the existence of $A\cup B$ without the union axiom?
negative_passages.docid: 3663
negative_passages.text: Finding counterexamples in elementary set theory.
negative_passages.docid: 63276
negative_passages.text: Is an arbitrary union of a chain of countable sets countable?
negative_passages.docid: 83185
negative_passages.text: How to show countability of $\omega^\omega$ or $\epsilon_0$ in ZF?
negative_passages.docid: 78290
negative_passages.text: Countable axiom of choice: why you can't prove it from just ZF
negative_passages.docid: 24050
negative_passages.text: Characterization of the Closure of a Set
negative_passages.docid: 49220
negative_passages.text: Open Sets of $\mathbb{R}^1$ and axiom of choice
negative_passages.docid: 29744
negative_passages.text: Compactness and axiomatisability
negative_passages.docid: 18223
negative_passages.text: Prove that the union of two disjoint countable sets is countable
negative_passages.docid: 12404
negative_passages.text: Union of definable sets
negative_passages.docid: 115675
negative_passages.text: Strange consequences of Axiom of Choice in Zermelo set theory
negative_passages.docid: 59820
negative_passages.text: ZF Extensionality axiom
negative_passages.docid: 88099
negative_passages.text: Not $\sigma$-compact set without axiom of choice
negative_passages.docid: 146664
negative_passages.text: The union of two linearly independent subsets
negative_passages.docid: 100726
negative_passages.text: Prove that the set of powers of a set, $\{ A^n : n \in \mathbb{N} \}$ exists by using ZFC axioms (without replacement).
negative_passages.docid: 77548
negative_passages.text: Given an arbitrary infinite set $S$, how to construct an injection $\mathbb{N} \to S$ with the Axiom of Choice?
negative_passages.docid: 127575
negative_passages.text: Prove that Open Sets in $\mathbb{R}$ are The Disjoint Union of Open Intervals Without the Axioms of Choice