query: "Weaker continuum hypothesis:" $2^{\aleph_0} < 2^{\aleph_1}$?
positive_passages.docid: 146432
positive_passages.text: If $2^{\aleph_0}=\aleph_2$, what is known about $2^{\aleph_1}$?
negative_passages.docid: 150605
negative_passages.text: Existence of a regular uncountable $\aleph_{\alpha}$ without $\mathsf{AC}$
negative_passages.docid: 122911
negative_passages.text: Characterization of weak convergence in $\ell_\infty$
negative_passages.docid: 4782
negative_passages.text: Can we distinguish $\aleph_0$ from $\aleph_1$ in Nature?
negative_passages.docid: 60004
negative_passages.text: Why is $\omega$ the smallest $\infty$?
negative_passages.docid: 24688
negative_passages.text: Why are $i$ and $-i$ "more indistinguishable" than $\sqrt{2}$ and $-\sqrt{2}$?
negative_passages.docid: 90537
negative_passages.text: Converse of Fermat's Little Theorem.
negative_passages.docid: 135524
negative_passages.text: Is $\mathrm{ZFC}^E$ outright inconsistent?
negative_passages.docid: 15416
negative_passages.text: A weak converse of $AB=BA\implies e^Ae^B=e^Be^A$ from "Topics in Matrix Analysis" for matrices of algebraic numbers.
negative_passages.docid: 65371
negative_passages.text: Does $2^X \cong 2^Y$ imply $X \cong Y$ without assuming the axiom of choice?
negative_passages.docid: 82464
negative_passages.text: $U^*\otimes V$ versus $L(U,V)$ for infinite dimensional spaces
negative_passages.docid: 76240
negative_passages.text: How is $\epsilon_0$ countable?
negative_passages.docid: 150937
negative_passages.text: Basic Algebra Proof on Integers - Weak Inequalities Work but Strict Inequalities Don't?
negative_passages.docid: 68995
negative_passages.text: Eisenstein's Criterion stronger version
negative_passages.docid: 132382
negative_passages.text: Showing a set is nowhere dense in $C([0,1])$
negative_passages.docid: 116811
negative_passages.text: Proofs with limit superior and limit inferior: $\liminf a_n \leq \limsup a_n$
negative_passages.docid: 149095
negative_passages.text: $f(\mathbb{R}\setminus \mathbb{Q}) \subseteq \mathbb{Q}$ and $f(\mathbb{Q}) \subseteq \mathbb{R}\setminus \mathbb{Q}$ imply that $f$ is not continuous