ⓘ Huge cardinal
In mathematics, a cardinal number κ is called huge if there exists an elementary embedding j: V → M from V into a transitive inner model M with critical point κ and
j κ M ⊂ M. {\displaystyle {}^{j\kappa}M\subset M.\!}Here, α M is the class of all sequences of length α whose elements are in M.
Huge cardinals were introduced by Kenneth Kunen 1978.
1. Variants
In what follows, j n refers to the n th iterate of the elementary embedding j, that is, j composed with itself n times, for a finite ordinal n. Also,
 Huge may refer to: Huge cardinal a number in mathematics Huge Caroline s Spine album 1996 Huge Hugh Hopper and Kramer album 1997 Huge TV series
 the size of the smallest witness to a large cardinal axiom. For example, the existence of a huge cardinal is much stronger, in terms of consistency strength
 n 1  superstrong cardinal exceeds that of an n  huge cardinal for each n 0. Kanamori, Akihiro 2003 The Higher Infinite : Large Cardinals in Set Theory
 cardinals η  extendible, extendible cardinals Vopenka cardinals Shelah for supercompactness, high jump cardinals n  superstrong n 2 n  almost huge
 I0 cardinal κ speaking here of the critical point of j is an I1 cardinal Every I1 cardinal κ sometimes called ω  huge cardinals is an I2 cardinal and
 measurable and also has κ  many measurable cardinals below it. Every measurable cardinal κ is a 0  huge cardinal because κM M, that is, every function from
 principle implies existence of Σn correct extendible cardinals for every n. If κ is an almost huge cardinal then a strong form of Vopenka s principle holds
 Ramsey cardinal Erdos cardinal Extendible cardinal Huge cardinal Hyper  Woodin cardinal Inaccessible cardinal Ineffable cardinal Mahlo cardinal Measurable
 contain the set j λ the image of j restricted to λ There is no ω  huge cardinal There is no non  trivial elementary embedding of Vλ 2 into itself. It
 Walter Kasper born 5 March 1933 is a German Roman Catholic Cardinal and theologian. He is President Emeritus of the Pontifical Council for Promoting

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