query: Proof that an infinite set can be written as the countably infinite union of pairwise disjoint infinite subsets without using the axiom of choice.
positive_passages.docid: 24372
positive_passages.text: Infinite Set is Disjoint Union of Two Infinite Sets
negative_passages.docid: 48523
negative_passages.text: Let $A$ be any uncountable set, and let $B$ be a countable subset of $A$. Prove that the cardinality of $A = A - B $
negative_passages.docid: 44974
negative_passages.text: Infinite set and countable subsets
negative_passages.docid: 128437
negative_passages.text: How to prove that a finite set is countable
negative_passages.docid: 6478
negative_passages.text: Prove that for any infinite poset there is an infinite subset which is either linearly ordered or antichain.
negative_passages.docid: 70940
negative_passages.text: Homework - set theory infinite union
negative_passages.docid: 151295
negative_passages.text: Proving $|S \times \mathbb{N}|=|S|$ when $S$ is an infinite set
negative_passages.docid: 21892
negative_passages.text: Any open subset of $\Bbb R$ is a countable union of disjoint open intervals
negative_passages.docid: 18223
negative_passages.text: Prove that the union of two disjoint countable sets is countable
negative_passages.docid: 91237
negative_passages.text: Every infinite set contains a countable subset
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: 91163
negative_passages.text: Let the set S be infinite, and the set T countably infinite. Show that S and S U T have the same cardinality
negative_passages.docid: 32774
negative_passages.text: Finite Cartesian Product of Countable sets is countable?
negative_passages.docid: 93419
negative_passages.text: Let $\mathcal A$ be a collection of pairwise disjoint subsets of a $\sigma$-finite measure space,Show that $\mathcal A$ is at most countable.
negative_passages.docid: 83842
negative_passages.text: Alternate proofs (other than diagonalization and topological nested sets) for uncountability of the reals?
negative_passages.docid: 9651
negative_passages.text: Cardinality of cartesian product of an infinite set with N
negative_passages.docid: 72750
negative_passages.text: Uncountable number of mutually disjoint countable sets in $\mathbb R$