query: Does $\operatorname{card} \operatorname{ran} f \leq \operatorname{card} \operatorname{dom} f$ imply choice?
positive_passages.docid: 88566
positive_passages.text: Is denying "Paradoxical Partitioning" equivalent to accepting the Axiom of Choice?
negative_passages.docid: 32180
negative_passages.text: Why is "$A$ unless $B$" equivalent to $A \lor B$?
negative_passages.docid: 105815
negative_passages.text: Does proving (second countable) $\Rightarrow$ (Lindelöf) require the axiom of choice?
negative_passages.docid: 115100
negative_passages.text: Operator Ordering Ambiguities
negative_passages.docid: 114190
negative_passages.text: If $X$ is separated over $\operatorname{Spec} \Bbb Z$, is it separated over any other scheme?
negative_passages.docid: 48088
negative_passages.text: Can the "Sacrifice (this card):" activated ability be placed back on the stack?
negative_passages.docid: 103427
negative_passages.text: When is $\operatorname{Hom}_A(M,N)$ finitely generated?
negative_passages.docid: 131592
negative_passages.text: Finite choice without AC
negative_passages.docid: 3902
negative_passages.text: Does $k+\aleph_0=\mathfrak{c}$ imply $k=\mathfrak{c}$ without the Axiom of Choice?
negative_passages.docid: 123447
negative_passages.text: Does a card have a keyword if it has the same effect as said keyword?
negative_passages.docid: 157728
negative_passages.text: When can you interchange composition of a limit?
negative_passages.docid: 41365
negative_passages.text: Choice function for collection of arbitrary finite sets. AC required?
negative_passages.docid: 135780
negative_passages.text: When does Slot evaluate?
negative_passages.docid: 2342
negative_passages.text: How does equality $\mathfrak{m}^\mathfrak{m} = 2^\mathfrak{m}$ in $\sf{ZF}$ relate to the axiom of choice?
negative_passages.docid: 119684
negative_passages.text: Is a contraction idempotent operator self-adjoint?
negative_passages.docid: 8342
negative_passages.text: Is short-circuiting logical operators mandated? And evaluation order?
negative_passages.docid: 98612
negative_passages.text: Is it possible to implement an equivalent syntax of (*x {\f(x)}) but the ordinate is specified first?