huge cardinal

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, is the class of all sequences of length less than $\backslash alpha$ whose elements are in $M$. Notice that for the "super" versions, $\backslash gamma$ should be less than $j(\backslash kappa)$, not $\{j^n(\backslash kappa)\}$.

κ is almost n-huge if and only if there is $j\; :\; V\; \backslash to\; M$ with critical point $\backslash kappa$ and

:

κ is super almost n-huge if and only if for every ordinal γ there is $j\; :\; V\; \backslash to\; M$ with critical point $\backslash kappa$, , and

:

κ is n-huge if and only if there is $j\; :\; V\; \backslash to\; M$ with critical point $\backslash kappa$ and

:$\{\}^\{j^n(\backslash kappa)\}M\; \backslash subset\; M.$

κ is super n-huge if and only if for every ordinal $\backslash gamma$ there is $j\; :\; V\; \backslash to\; M$ with critical point $\backslash kappa$, , and

:$\{\}^\{j^n(\backslash kappa)\}M\; \backslash subset\; M.$

Notice that 0-huge is the same as measurable cardinal; and 1-huge is the same as huge. A cardinal satisfying one of the rank into rank axioms is $n$-huge for all finite $n$.

The existence of an almost huge cardinal implies that Vopěnka's principle is consistent; more precisely any almost huge cardinal is also a Vopěnka cardinal.

Kanamori, Reinhardt, and Solovay defined seven large cardinal properties between extendibility and hugeness in strength, named $\backslash mathbf\; A\_2(\backslash kappa)$ through $\backslash mathbf\; A\_7(\backslash kappa)$, and a property $\backslash mathbf\; A\_6^\backslash ast(\backslash kappa)$.A. Kanamori, W. N. Reinhardt, R. Solovay, "[https://math.bu.edu/people/aki/d.pdf Strong Axioms of Infinity and Elementary Embeddings]", pp.110--111. Annals of Mathematical Logic vol. 13 (1978). The additional property $\backslash mathbf\; A\_1(\backslash kappa)$ is equivalent to "$\backslash kappa$ is huge", and $\backslash mathbf\; A\_3(\backslash kappa)$ is equivalent to "$\backslash kappa$ is $\backslash lambda$-supercompact for all

The cardinals are arranged in order of increasing consistency strength as follows:

• almost $n$-huge
• super almost $n$-huge
• $n$-huge
• super $n$-huge
• almost $n+1$-huge

The consistency of a huge cardinal implies the consistency of a supercompact cardinal, nevertheless, the least huge cardinal is smaller than the least supercompact cardinal (assuming both exist).

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One can try defining an $\backslash omega$-huge cardinal $\backslash kappa$ as one such that an elementary embedding $j\; :\; V\; \backslash to\; M$ from $V$ into a transitive inner model $M$ with critical point $\backslash kappa$ and $\{\}^\backslash lambda\; M\backslash subseteq\; M$, where $\backslash lambda$ is the supremum of $j^n(\backslash kappa)$ for positive integers $n$. However Kunen's inconsistency theorem shows that such cardinals are inconsistent in ZFC, though it is still open whether they are consistent in ZF. Instead an $\backslash omega$-huge cardinal $\backslash kappa$ is defined as the critical point of an elementary embedding from some rank $V\_\{\backslash lambda+1\}$ to itself. This is closely related to the rank-into-rank axiom I₁.

• List of large cardinal properties
• The Dehornoy order on a braid group was motivated by properties of huge cardinals.

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• {{Citation|last=Kanamori|first=Akihiro|authorlink=Akihiro Kanamori|year=2003|publisher=Springer|title=The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings|title-link= The Higher Infinite |edition=2nd|isbn=3-540-00384-3}}.
• {{Citation | last=Kunen | first=Kenneth | authorlink=Kenneth Kunen | title=Saturated ideals | doi=10.2307/2271949 | jstor=2271949 | mr=495118 | year=1978 | journal=The Journal of Symbolic Logic | issn=0022-4812 | volume=43 | issue=1 | pages=65–76| s2cid=13379542 }}.
• {{Citation|last=Maddy|first=Penelope|authorlink=Penelope Maddy|journal=The Journal of Symbolic Logic|title=Believing the Axioms. II|year=1988|volume=53|issue=3|pages=736-764 (esp. 754-756)|doi=10.2307/2274569|jstor=2274569|s2cid=16544090 }}. A copy of parts I and II of this article with corrections is available at the [http://faculty.sites.uci.edu/pjmaddy/bibliography/ author's web page].

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Category:Large cardinals