Cantor–Bernstein theorem
There are equally many countable order types and real numbers
In set theory and order theory, the Cantor–Bernstein theorem states that the cardinality of the second type class, the class of countable order types, equals the cardinality of the continuum. It was used by Felix Hausdorff and named by him after Georg Cantor and Felix Bernstein. Cantor constructed a family of countable order types with the cardinality of the continuum, and in his 1901 inaugural dissertation Bernstein proved that such a family can have no higher cardinality.[1]
References
- ^ Plotkin, J. M., ed. (2005). Hausdorff on Ordered Sets. History of Mathematics. Vol. 25. American Mathematical Society. p. 3. ISBN 9780821890516..
- v
- t
- e
Order theory
- Binary relation
- Boolean algebra
- Cyclic order
- Lattice
- Partial order
- Preorder
- Total order
- Weak ordering
- Antisymmetric
- Asymmetric
- Boolean algebra
- Completeness
- Connected
- Covering
- Dense
- Directed
- (Partial) Equivalence
- Foundational
- Heyting algebra
- Homogeneous
- Idempotent
- Lattice
- Reflexive
- Partial order
- Prefix order
- Preorder
- Semilattice
- Semiorder
- Symmetric
- Total
- Tolerance
- Transitive
- Well-founded
- Well-quasi-ordering (Better)
- (Pre) Well-order
- Alexandrov topology & Specialization preorder
- Ordered topological vector space
- Normal cone
- Order topology
- Order topology
- Topological vector lattice
- Antichain
- Cofinal
- Cofinality
- Comparability
- Duality
- Filter
- Hasse diagram
- Ideal
- Net
- Subnet
- Order morphism
- Order type
- Ordered field
- Ordered vector space
- Partially ordered group
- Upper set
- Young's lattice