query: Proof that the finite product of nonempty sets is nonempty without axiom of choice from ZF
positive_passages.docid: 131592
positive_passages.text: Finite choice without AC
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
negative_passages.docid: 88736
negative_passages.text: How does the Axiom of Extensionality prove the uniqueness of Specified sets?
negative_passages.docid: 154578
negative_passages.text: Proving there is a sequence convergent to a limit point of a set without axiom of countable choice?
negative_passages.docid: 27899
negative_passages.text: Can we prove the existence of $A\cup B$ without the union axiom?
negative_passages.docid: 272
negative_passages.text: Cantor Sets/nonempty/cardinality
negative_passages.docid: 49621
negative_passages.text: Proof of "well-founded iff there is no sequence $x_n$ s.t. $(x_{n+1}, x_n) \in R$"
negative_passages.docid: 59820
negative_passages.text: ZF Extensionality axiom
negative_passages.docid: 41783
negative_passages.text: Cardinality of the Cartesian Product of Two Equinumerous Infinite Sets
negative_passages.docid: 31389
negative_passages.text: consistency of ZF\{Power Set axiom}
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: 12436
negative_passages.text: The cardinality of the set of all finite subsets of an infinite set
negative_passages.docid: 32774
negative_passages.text: Finite Cartesian Product of Countable sets is countable?
negative_passages.docid: 142219
negative_passages.text: Does the splitting lemma hold without the axiom of choice?
negative_passages.docid: 101453
negative_passages.text: Existence of a Boolean algebra with a unique ultrafilter in ZF
negative_passages.docid: 139992
negative_passages.text: Is power-associativity finitely axiomatizable?
negative_passages.docid: 144899
negative_passages.text: Do We Need the Axiom of Choice for Finite Sets?