query: Why is the $\aleph$ hierarchy a discrete hierarchy?
positive_passages.docid: 151355
positive_passages.text: Are cardinal numbers well defined? Or could we have something like $\aleph_{1/2}$?
negative_passages.docid: 150605
negative_passages.text: Existence of a regular uncountable $\aleph_{\alpha}$ without $\mathsf{AC}$
negative_passages.docid: 72190
negative_passages.text: Topological proof on discrete topology where $X$ is infinite
negative_passages.docid: 62977
negative_passages.text: Show on $\mathbb{N}$ there are $2^{\aleph_0}$ nonisomorphic linear orders.
negative_passages.docid: 16534
negative_passages.text: Can $(\Bbb N,\leq)$ have an $\aleph_0$-categorical theory (in a larger language)?
negative_passages.docid: 23359
negative_passages.text: Does the Planck scale imply that spacetime is discrete?
negative_passages.docid: 16656
negative_passages.text: Atiyah and Macdonald, Proposition 2.9
negative_passages.docid: 57398
negative_passages.text: Where does TREE(n) sit on the Fast Growing Hierarchy?
negative_passages.docid: 40937
negative_passages.text: Is the set of partial order relations over $ \mathbb{N}$ countable?
negative_passages.docid: 70785
negative_passages.text: A finite Hausdorff space is discrete
negative_passages.docid: 91545
negative_passages.text: A stronger version of discrete "Liouville's theorem"
negative_passages.docid: 146432
negative_passages.text: If $2^{\aleph_0}=\aleph_2$, what is known about $2^{\aleph_1}$?
negative_passages.docid: 6681
negative_passages.text: How does Cantor's diagonal argument work?
negative_passages.docid: 137075
negative_passages.text: If a topology contains all infinite subsets, then it is the discrete topology
negative_passages.docid: 91756
negative_passages.text: Degeneracy in one dimension
negative_passages.docid: 5035
negative_passages.text: Can unprovability unprovable? Is there an $\omega$-fold unprovability?
negative_passages.docid: 56528
negative_passages.text: On the proof of existence and uniqueness of radix representation