Untersuchungen aus der Mengenlehre¶
Bernstein, F. (1898). Untersuchungen aus der Mengenlehre. Mathematische Annalen.
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Primes¶
- Cardinality
- applicable to finite and infinite sets uniformly, and is itself an equivalence relation (reflexive via identity, symmetric via inverse, transitive via composition); cardinal numbers are the equivalence classes under this equinumerosity, and the relation \(A \approx B\) is sometimes called equipollence; (2) cardinalities admit a natural ordering:
This sourceSupplies the antisymmetry of the cardinal order: if |A| ≤ |B| and |B| ≤ |A| then |A| = |B|, with a proof that does not depend on the Axiom of Choice. (Cantor stated the theorem in 1895; Bernstein proved it 1897, published 1898 via Borel; Schröder's contemporaneous 1898 proof was flawed; Dedekind had an earlier unpublished proof.)
- applicable to finite and infinite sets uniformly, and is itself an equivalence relation (reflexive via identity, symmetric via inverse, transitive via composition); cardinal numbers are the equivalence classes under this equinumerosity, and the relation \(A \approx B\) is sometimes called equipollence; (2) cardinalities admit a natural ordering:
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