Difference between revisions of "Huge"

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Where $ot(X)$ is the [[Order-isomorphism|order-type]] of the poset $(X,\in)$. <cite>Kanamori2009:HigherInfinite</cite> This definition is, more intuitively, making $U$ very large, like most ultrafilter characterizations of large cardinals ([[supercompact]], [[strongly compact]], etc.).
 
Where $ot(X)$ is the [[Order-isomorphism|order-type]] of the poset $(X,\in)$. <cite>Kanamori2009:HigherInfinite</cite> This definition is, more intuitively, making $U$ very large, like most ultrafilter characterizations of large cardinals ([[supercompact]], [[strongly compact]], etc.).
 
$\kappa$ is then super $n$-huge if for all ordinals $\theta$ there is a $\lambda>\theta$ such that $\kappa$ is $n$-huge with target $\lambda$, i.e. $\lambda_n$ can be made arbitrarily large. <cite>Kanamori2009:HigherInfinite</cite>
 
$\kappa$ is then super $n$-huge if for all ordinals $\theta$ there is a $\lambda>\theta$ such that $\kappa$ is $n$-huge with target $\lambda$, i.e. $\lambda_n$ can be made arbitrarily large. <cite>Kanamori2009:HigherInfinite</cite>
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If $j:V\to M$ is such that $M^{j^n(\kappa)}\subseteq M$ (i.e. $j$ witnesses $n$-hugeness) then there is a ultrafilter $U$ as above such that, for all $k\leq n$, $\lambda_k = j^k(\kappa)$, i.e. it is not only $\lambda=\lambda_n$ that is an iterate of $\kappa$ by $j$; all members of the $\lambda_k$ sequence are.
  
 
== Consistency strength and size ==
 
== Consistency strength and size ==

Revision as of 14:21, 11 November 2017

Huge cardinals (and their variants) were introduced by Kenneth Kunen in 1972 as a very large cardinal axiom. Kenneth Kunen first used them to prove that the consistency of the existence of a huge cardinal implies the consistency of $\text{ZFC}$+"there is a $\aleph_2$-saturated ideal over $\omega_1$". [1]

Definitions

Their formulation is similar to that of the formulation of superstrong cardinals. A huge cardinal is to a supercompact cardinal as a superstrong cardinal is to a strong cardinal, more precisely. The definition is part of a generalized phenomenon known as the "double helix", in which for some large cardinal properties $n$-$P_0$ and $n$-$P_1$, $n$-$P_0$ has less consistency strength than $n$-$P_1$, which has less consistency strength than $(n+1)$-$P_0$, and so on. This phenomenon is seen only around the $n$-fold variants as of modern set theoretic concerns. [2]

Although they are very large, there is a first-order definition which is equivalent to $n$-hugeness, so the $\theta$-th $n$-huge cardinal is first-order definable whenever $\theta$ is first-order definable. This definition can be seen as a (very strong) strengthening of the first-order definition of measurability.

Elementary embedding definitions

The elementary embedding definitions are somewhat standard. Let $j:V\rightarrow M$ be an elementary embedding with critical point $\kappa$ such that $M$ is a standard inner model of $\text{ZFC}$. Then:

  • $\kappa$ is almost $n$-huge with target $\lambda$ iff $\lambda=j^n(\kappa)$ and $M$ is closed under all of its sequences of length less than $\lambda$ (that is, $M^{<\lambda}\subset M$).
  • $\kappa$ is $n$-huge with target $\lambda$ iff $\lambda=j^n(\kappa)$ and $M$ is closed under all of its sequences of length $\lambda$ ($M^\lambda\subset M$).
  • $\kappa$ is almost $n$-huge iff it is almost $n$-huge with target $\lambda$ for some $\lambda$.
  • $\kappa$ is $n$-huge iff it is $n$-huge with target $\lambda$ for some $\lambda$.
  • $\kappa$ is super almost $n$-huge iff for every $\gamma$, there is some $\lambda>\gamma$ for which $\kappa$ is almost $n$-huge with target $\lambda$ (that is, the target can be made arbitrarily large).
  • $\kappa$ is super $n$-huge iff for every $\gamma$, there is some $\lambda>\gamma$ for which $\kappa$ is $n$-huge with target $\lambda$.
  • $\kappa$ is almost huge, huge, super almost huge, and superhuge iff it is almost $1$-huge, $1$-huge, etc. respectively.

Ultrafilter definition

The first-order definition of $n$-huge is somewhat similar to measurability. Specifically, $\kappa$ is measurable iff there is a nonprincipal $\kappa$-complete ultrafilter, $U$, over $\kappa$. A cardinal $\kappa$ is $n$-huge with target $\lambda$ iff there is a normal $\kappa$-complete ultrafilter, $U$, over $\mathcal{P}(\lambda)$, and cardinals $\kappa=\lambda_0<\lambda_1<\lambda_2...<\lambda_{n-1}<\lambda_n=\lambda$ such that:

$$\forall i<n\forall x\subseteq\lambda(ot(x\cap\lambda_{i+1})=\lambda_i\rightarrow x\in U)$$

Where $ot(X)$ is the order-type of the poset $(X,\in)$. [1] This definition is, more intuitively, making $U$ very large, like most ultrafilter characterizations of large cardinals (supercompact, strongly compact, etc.). $\kappa$ is then super $n$-huge if for all ordinals $\theta$ there is a $\lambda>\theta$ such that $\kappa$ is $n$-huge with target $\lambda$, i.e. $\lambda_n$ can be made arbitrarily large. [1]

If $j:V\to M$ is such that $M^{j^n(\kappa)}\subseteq M$ (i.e. $j$ witnesses $n$-hugeness) then there is a ultrafilter $U$ as above such that, for all $k\leq n$, $\lambda_k = j^k(\kappa)$, i.e. it is not only $\lambda=\lambda_n$ that is an iterate of $\kappa$ by $j$; all members of the $\lambda_k$ sequence are.

Consistency strength and size

Hugeness exhibits a phenomenon associated with similarly defined large cardinals (the $n$-fold variants) known as the double helix. This phenomenon is when for one $n$-fold variant, letting a cardinal be called $n$-$P_0$ iff it has the property, and another variant, $n$-$P_1$, $n$-$P_0$ is weaker than $n$-$P_1$, which is weaker than $(n+1)$-$P_0$. [2] In the consistency strength hierarchy, here is where these lay (top being weakest):

  • measurable = $0$-superstrong = almost $0$-huge = super almost $0$-huge = $0$-huge = super $0$-huge
  • $n$-superstrong
  • $n$-fold supercompact
  • $(n+1)$-fold strong, $n$-fold extendible
  • $(n+1)$-fold Woodin, $n$-fold Vopěnka
  • $(n+1)$-fold Shelah
  • almost $n$-huge
  • super almost $n$-huge
  • $n$-huge
  • super $n$-huge
  • $(n+1)$-superstrong

All huge variants lay at the top of the double helix restricted to some natural number $n$, although each are bested by I3 cardinals (the critical points of the I3 elementary embeddings). In fact, every I3 is preceeded by a stationary set of $n$-huge cardinals, for all $n$. [1]

Similarly, every huge cardinal $\kappa$ is almost huge, and there is a normal measure over $\kappa$ which contains every almost huge cardinal $\lambda<\kappa$. Every superhuge cardinal $\kappa$ is extendible and there is a normal measure over $\kappa$ which contains every extendible cardinal $\lambda<\kappa$. Every $(n+1)$-huge cardinal $\kappa$ has a normal measure which contains every cardinal $\lambda$ such that $V_\kappa\models$"$\lambda$ is super $n$-huge". [1]

In terms of size, however, the least $n$-huge cardinal is smaller than the least supercompact cardinal. Assuming both exist, for any $\kappa$ which is supercompact and has an $n$-huge cardinal above it, there are $\kappa$ many $n$-huge cardinals less than $\kappa$. [1]

Every $n$-huge cardinal is $m$-huge for every $m\leq n$. Similarly with almost $n$-hugeness, super $n$-hugeness, and super almost $n$-hugeness. Every almost huge cardinal is Vopěnka (therefore the consistency of the existence of an almost-huge cardinal implies the consistency of Vopěnka's principle). [1]

References

  1. Kanamori, Akihiro. The higher infinite. Second, Springer-Verlag, Berlin, 2009. (Large cardinals in set theory from their beginnings, Paperback reprint of the 2003 edition) www   bibtex
  2. Kentaro, Sato. Double helix in large large cardinals and iteration of elementary embeddings. Annals of Pure and Applied Logic 146(2-3):199-236, May, 2007. www   DOI   bibtex
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