query: Counterexample: a pair of linearly ordered sets that are isomorphic to subsets of the other, but not isomorphic between them
positive_passages.docid: 146062
positive_passages.text: A "Cantor-Schroder-Bernstein" theorem for partially-ordered-sets
negative_passages.docid: 98391
negative_passages.text: example of a nonempty subset is closed under scalar multiplication but not a subspace
negative_passages.docid: 154230
negative_passages.text: In $\mathsf{ZF}$, for each set $A$ exists an ordinal $\alpha$ such that no surjection $A\to \alpha$ exists
negative_passages.docid: 141318
negative_passages.text: $\subset$ vs $\subseteq$ when *not* referring to strict inclusion
negative_passages.docid: 115649
negative_passages.text: Proving two sets $A, B$ are order isomorphic
negative_passages.docid: 13779
negative_passages.text: Canonical copies of sets
negative_passages.docid: 88099
negative_passages.text: Not $\sigma$-compact set without axiom of choice
negative_passages.docid: 1508
negative_passages.text: How do I prove that a definition of an ordered pair does not satisfy the required property?
negative_passages.docid: 45878
negative_passages.text: Elementary Set Theory: Ordinals, Well Orderings and Isomorphisms
negative_passages.docid: 78300
negative_passages.text: Showing a set is closed other than giving a counterexample?
negative_passages.docid: 83996
negative_passages.text: For any two sets $A,B$ , $|A|\leq|B|$ or $|B|\leq|A|$
negative_passages.docid: 60306
negative_passages.text: Natural class of functions whose $\mathcal O$-sets are linearly ordered
negative_passages.docid: 122271
negative_passages.text: A subset of a transitive set is transitive
negative_passages.docid: 112231
negative_passages.text: Showing that a one-to-one linear transformation maps a linearly independent set onto a linearly independent set
negative_passages.docid: 12404
negative_passages.text: Union of definable sets
negative_passages.docid: 29755
negative_passages.text: Power set equivalency of Axiom of Choice
negative_passages.docid: 132868
negative_passages.text: Bijection between an infinite set and its union of a countably infinite set