query: On finding enough rationals (countable) to fill the uncountable number of intervals between the irrationals.
positive_passages.docid: 34825
positive_passages.text: Intuitive explanation for how could there be "more" irrational numbers than rational?
negative_passages.docid: 63804
negative_passages.text: Proof that every repeating decimal is rational
negative_passages.docid: 64885
negative_passages.text: Irrational and rational sequence proof
negative_passages.docid: 68456
negative_passages.text: Prove that there exists irrational numbers $x$ and $y$ such that $x + y$ is rational, without using subtraction
negative_passages.docid: 139826
negative_passages.text: arbitrary large finite sums of an uncountable set.
negative_passages.docid: 34710
negative_passages.text: Cardinality of Irrational Numbers
negative_passages.docid: 83842
negative_passages.text: Alternate proofs (other than diagonalization and topological nested sets) for uncountability of the reals?
negative_passages.docid: 34315
negative_passages.text: Rational, integers and probability
negative_passages.docid: 77129
negative_passages.text: Density of irrationals
negative_passages.docid: 68230
negative_passages.text: The cardinality of a countable union of sets with less than continuum cardinality
negative_passages.docid: 23752
negative_passages.text: Proving Irrationality
negative_passages.docid: 149317
negative_passages.text: An open subset of $\mathbb{R}$ is an at-most-countable union of disjoint open intervals
negative_passages.docid: 130361
negative_passages.text: Basis for a topology with a countable number of sets
negative_passages.docid: 114442
negative_passages.text: Completion of rational numbers via Cauchy sequences
negative_passages.docid: 4381
negative_passages.text: Proof that the set of irrational numbers is dense in reals
negative_passages.docid: 101028
negative_passages.text: Irrational numbers and Borel Sets
negative_passages.docid: 130981
negative_passages.text: Co-countable measure on uncountable set