query: Proof that $\aleph_1$ exists.
positive_passages.docid: 28764
positive_passages.text: How do we know an $ \aleph_1 $ exists at all?
negative_passages.docid: 130916
negative_passages.text: Proving the identity functions is continuous iff $\tau_1\supseteq\tau_2$
negative_passages.docid: 24785
negative_passages.text: How prove this equation $\alpha=\aleph_{0}\alpha$
negative_passages.docid: 150385
negative_passages.text: The use of the Axiom of Choice in this proof of the sequence lemma
negative_passages.docid: 99782
negative_passages.text: $a_{n+1}=|a_n|-a_{n-1} \implies a_n \; \text{is periodic}$
negative_passages.docid: 145752
negative_passages.text: Aut $\mathbb Z_p\simeq \mathbb Z_{p-1}$
negative_passages.docid: 34154
negative_passages.text: Prove the statement $\forall x\in\mathbb N (x > 1\to\exists k\in\mathbb N\exists m \in\mathbb N (m \equiv 1 (\text{mod }2) \wedge x = 2^km))$.
negative_passages.docid: 158257
negative_passages.text: Is aleph-$0$ a natural number?
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: 134720
negative_passages.text: Show that there is no increasing function $f: \aleph_1 \to \mathbb{R}$
negative_passages.docid: 84021
negative_passages.text: How can we show that $\varphi(a) \neq a +1$ for every $a \in A$?
negative_passages.docid: 30770
negative_passages.text: Prove that $C\left(\bigcup_{\alpha\in I} A_\alpha\right)=\bigcap_{\alpha\in I}C\left(A_\alpha\right)$
negative_passages.docid: 94383
negative_passages.text: Prove that $A^TD-C^TB=I$
negative_passages.docid: 146432
negative_passages.text: If $2^{\aleph_0}=\aleph_2$, what is known about $2^{\aleph_1}$?
negative_passages.docid: 17191
negative_passages.text: How to verify $H\otimes K \cong \bigoplus\limits_{i\in I}H$
negative_passages.docid: 67066
negative_passages.text: Proof there is a 1-1 correspondence between an uncountable set and itself minus a countable part of it
negative_passages.docid: 58186
negative_passages.text: Set theory proof, $\exists g(g\colon A\twoheadrightarrow C)$ or $\exists h(h\colon C\hookrightarrow B)$