Hartogs number

English Alternative forms en Hartogs' number Etymology After German-Jewish mathematician Friedrich Hartogs (1874–1943). Noun en set theory For a given set X, the cardinality of the least ordinal number α such that there is no injection from α into X. 1973 [North-Holland], Thomas J. Jech, The Axiom of Choice, 2013, Dover, page 160, Let be an infinite cardinal, and let be the Hartogs number of . 1995, The Bulletin of Symbolic Logic, Volume 1, Association for Symbolic Logic, page 139, If the Power Set Axiom is replaced by " is bound for every x" where is one-to-one function from into , then the theory is denoted by ZFH (H stands for Hartogs' Number). en 2014 Barnaby Sheppard The Logic of Infinity https://books.google.com.au/books?id=mxZEBAAAQBAJ&pg=PA341&dq=%22Hartogs%27s+number%22%7C%22Hartogs%27+number%22&hl=en&sa=X&ved=0ahUKEwi_0aXO8tXcAhXklFQKHQ0FDJAQ6AEIPjAD#v=onepage&q=%22Hartogs's%20number%22%7C%22Hartogs'%20number%22&f=false 341 w:Cambridge University Press 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. Usage notes The Hartogs number is a cardinal number representing the size of the ordinal number α (regarded as a set). The definition is worded such that X does not need to have a well-order. Translations cardinality of the least ordinal for which there is no injection into a given set See also en equipotent en Hartogs function en eponyms