query: Are there sets $S\subseteq\Bbb N$ which are provably non-empty, but we don't know what is $\min S$?
positive_passages.docid: 143274
positive_passages.text: What is the smallest unknown natural number?
negative_passages.docid: 45405
negative_passages.text: An elementary way to show any bounded subset of $\Bbb{R}^k$ is totally bounded
negative_passages.docid: 858
negative_passages.text: Is $\{\emptyset\}$ a subset of $\{\{\emptyset\}\}$?
negative_passages.docid: 88099
negative_passages.text: Not $\sigma$-compact set without axiom of choice
negative_passages.docid: 78117
negative_passages.text: Closed property of nonempty finite set
negative_passages.docid: 132692
negative_passages.text: Nonempty set mapped to $\emptyset$ and vice-versa
negative_passages.docid: 121097
negative_passages.text: Every not empty finite subset of a totally ordered set has a maximum and minimum
negative_passages.docid: 67259
negative_passages.text: Can we prove that a bounded closed subset of $\mathbb R^n$ is compact without Axiom of Choice?
negative_passages.docid: 55995
negative_passages.text: Why is the supremum of the empty set $-\infty$ and the infimum $\infty$?
negative_passages.docid: 27889
negative_passages.text: If the set of odd numbers is a subset of $\mathbb N$ then surely it is smaller than $\mathbb N$
negative_passages.docid: 65549
negative_passages.text: What is the cartesian product of an empty set of sets?
negative_passages.docid: 11817
negative_passages.text: Proving that every set $A \subset \Bbb N$ of size $n$ contains a subset $B \subset A$ with $n | \sum_{b \in B} b$
negative_passages.docid: 130771
negative_passages.text: Sets and the principle of inclusion and exclusion
negative_passages.docid: 156006
negative_passages.text: Cardinality of set containing all infinite subsets of $\mathbb{Q}$
negative_passages.docid: 136755
negative_passages.text: Direct proof of empty set being subset of every set
negative_passages.docid: 103314
negative_passages.text: Let $\{K_n \}_{n \in \mathbb{N}}$ be a collection of sets, where $K_n = [n, \infty)$. Why is $\bigcap_{n \in \mathbb{N}} K_{n}$ empty?
negative_passages.docid: 86309
negative_passages.text: Whether the set $A$ is a compact subset of $M_3(\Bbb R)$.