Huge cardinal

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Huge cardinals (and their variants) were introduced by Kenneth Kunen in 1978 as a very large cardinal axiom. They were introduced in 1972 by Kenneth Kunen who proved that the consistency of the existence of a huge cardinal implied the consistency of $ZFC+$"there is a $\omega_2$-saturated ideal over $\omega_1$".

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. [1]

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 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 nonprinciple $\kappa$-complete ultrafilter, $U$, over $\kappa$. A cardinal $\kappa$ is $n$-huge iff there is some cardinal $\lambda$, a nonprinciple $\kappa$-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 $(X,\in)$. [2] This definition is, more intuitively, making $U$ very large, like most ultrafilter characterizations of large cardinals (supercompact, strongly compact, etc.).

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 $n+1$-$P_0$. [1] In the consistency strength hierarchy, here is where these lay (top being weakest):

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 $n$. [2]

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 ultrafilter which contains every cardinal $\lambda$ such that $V_\kappa\models\lambda\;\mathrm{is}\;\mathrm{super n-huge}$. [2]

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 $\kappa$. [2]

Every $n$-huge cardinal is $m$-huge for every $m\leq n$[2]. 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 a almost-huge cardinal implies the consistency of Vopěnka's principle). [2]

References

  1. Kentaro, Sato. Double helix in large large cardinals and iteration ofelementary embeddings. , 2007. www   bibtex
  2. 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
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