query: Impossibility of constructing a continuum-size linearly independent set in $\Bbb R$
positive_passages.docid: 146560
positive_passages.text: Is there any uncountably infinite set that does not generate the reals?
negative_passages.docid: 136106
negative_passages.text: Is every open subset of $ \mathbb{R} $ uncountable?
negative_passages.docid: 60424
negative_passages.text: Uncountable family of uncountable compact subsets of $\mathbb{R}$
negative_passages.docid: 22558
negative_passages.text: Vitali-type set with given outer measure
negative_passages.docid: 19414
negative_passages.text: Uncountable subset with uncountable complement, without the Axiom of Choice
negative_passages.docid: 126569
negative_passages.text: Can we find a well ordering on an infinite set with no largest element?
negative_passages.docid: 3023
negative_passages.text: Construction of a Borel set with positive but not full measure in each interval
negative_passages.docid: 12449
negative_passages.text: How to formulate continuum hypothesis without the axiom of choice?
negative_passages.docid: 72420
negative_passages.text: Is $\mathbb{N}$ impossible to pin down?
negative_passages.docid: 88099
negative_passages.text: Not $\sigma$-compact set without axiom of choice
negative_passages.docid: 5035
negative_passages.text: Can unprovability unprovable? Is there an $\omega$-fold unprovability?
negative_passages.docid: 127678
negative_passages.text: A partition of the unit interval into uncountably many dense uncountable subsets
negative_passages.docid: 132382
negative_passages.text: Showing a set is nowhere dense in $C([0,1])$
negative_passages.docid: 97064
negative_passages.text: Dense set in an infinite cofinite space
negative_passages.docid: 77548
negative_passages.text: Given an arbitrary infinite set $S$, how to construct an injection $\mathbb{N} \to S$ with the Axiom of Choice?
negative_passages.docid: 272
negative_passages.text: Cantor Sets/nonempty/cardinality
negative_passages.docid: 92249
negative_passages.text: Every finite set contains its supremum: proof improvement.