query: Proving that if a set $A$ is infinite then necessarily $|A|\geq|\mathbb{N}|$
positive_passages.docid: 60004
positive_passages.text: Why is $\omega$ the smallest $\infty$?
negative_passages.docid: 149673
negative_passages.text: Cardinality of an infinite union of finite sets
negative_passages.docid: 154559
negative_passages.text: For each $x\in X$ there exists a neighborhood of x s.t $A\cap U\neq \emptyset$ for only finitely many $A\in\mathcal{A}$. Show that f is continuous.
negative_passages.docid: 23442
negative_passages.text: Cardinality of the union of infinite and countable sets
negative_passages.docid: 119624
negative_passages.text: A Dedekind infinite set has a countably infinite proper subset
negative_passages.docid: 86926
negative_passages.text: Why isn't this a well ordering of $\{A\subseteq\mathbb N\mid A\text{ is infinite}\}$?
negative_passages.docid: 81682
negative_passages.text: Proving the algebraic set corresponding to a polynomial is infinite.
negative_passages.docid: 156006
negative_passages.text: Cardinality of set containing all infinite subsets of $\mathbb{Q}$
negative_passages.docid: 157513
negative_passages.text: How can I prove that these sets are countable infinite?
negative_passages.docid: 127242
negative_passages.text: $|A\times B|= \text{max}(|A|,|B|)$ for infinite sets
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: 70940
negative_passages.text: Homework - set theory infinite union
negative_passages.docid: 120562
negative_passages.text: If A is an infinite set and B is at most countable set, prove that A and $A \cup B$ have the same cardinality
negative_passages.docid: 91237
negative_passages.text: Every infinite set contains a countable subset
negative_passages.docid: 41661
negative_passages.text: Show that the set of all finite subsets of $\mathbb{N}$ is countable.
negative_passages.docid: 66301
negative_passages.text: The cardinality of the set of countably infinite subsets of an infinite set
negative_passages.docid: 102478
negative_passages.text: $X_n\leq Y_n$ implies $\liminf X_n \leq \liminf Y_n$ and $\limsup X_n \leq \limsup Y_n$