# ⓘ 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 c ..

## ⓘ 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,

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• cardinals η - extendible, extendible cardinals Vopenka cardinals Shelah for supercompactness, high jump cardinals n - superstrong n 2 n - almost huge
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• 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