Meaning of Hartogs number | Babel Free
Definitions
For a given set X, the cardinality of the least ordinal number α such that there is no injection from α into X.
Examples
“1973 [North-Holland], Thomas J. Jech, The Axiom of Choice, 2013, Dover, page 160, Let⟳ p be an infinite cardinal, |X|=p and let⟳ א=א(p) be the Hartogs number⟳ of p.”
“1995, The Bulletin of Symbolic Logic, Volume 1, Association for Symbolic Logic, page 139, If the Power Set⟳ Axiom is replaced by "א(x) is bound for every x" where א(x)={a|∃f(f is one-to-one function from a into x)}, then the theory is denoted by ZFH (H stands for Hartogs' Number⟳).”
“Since the proof of Hartogs' Theorem does not appeal to the Axiom of Choice, the Hartogs number⟳ of a set⟳ X exists whether or not X has a well-ordering.”
CEFR level
B2
Upper Intermediate
This word is part of the CEFR B2 vocabulary — upper intermediate level.
This word is part of the CEFR B2 vocabulary — upper intermediate level.
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