Hartogs number
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In mathematics, specifically in axiomatic set theory, a Hartogs number is a particular kind of cardinal number. It was shown by Friedrich Hartogs in 1915, from ZF alone (that is, without using the axiom of choice), that there is a least wellordered cardinal greater than a given wellordered cardinal.
To define the Hartogs number of a set it is not in fact necessary that the set be wellorderable: If X is any set, then the Hartogs number of X is the least ordinal α such that there is no injection from α into X. If X cannot be wellordered, then we can no longer say that this α is the least wellordered cardinal greater than the cardinality of X, but it remains the least wellordered cardinal not less than or equal to the cardinality of X.
[edit] References
- Hartogs, Friedrich (1915). "Über das Problem der Wohlordnung". Mathematische Annalen 76: 438-443.
- Jech, Thomas (2002). Set theory, third millennium edition (revised and expanded). Springer. ISBN 3-540-44085-2.