Difference between revisions of "Huge"

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*$\kappa$ is '''super $n$-huge''' iff for every $\gamma$, there is some $\lambda>\gamma$ for which $\kappa$ is $n$-huge with target $\lambda$.
 
*$\kappa$ is '''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.
 
*$\kappa$ is '''almost huge''', '''huge''', '''super almost huge''', and '''superhuge''' iff it is '''almost 1-huge''', '''1-huge''', etc. respectively.
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=== Ultrahuge cardinals, $C^{(n)}$-ultrahuge cardinals ===
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A cardinal $\kappa$ is '''$\lambda$-ultrahuge''' for $\lambda>\kappa$ if there exists a nontrivial elementary embedding $j:V\to M$ for some transitive class $M$ such that $\mathrm{j}(\kappa)>\lambda$, $M^{j(\kappa)}\subseteq M$ and $V_{j(\lambda)}\subseteq M$. A cardinal is '''ultrahuge''' if it is $\lambda$-ultrahuge for all $\lambda\geq\kappa$. [http://logicatorino.altervista.org/slides/150619tsaprounis.pdf] Notice how similar this definition is to the alternative characterization of [[extendible]] cardinals. Furthermore, this definition can be extended in the obvious way to define $\lambda$-ultra $n$-hugeness and ultra $n$-hugeness, as well as the "''almost''" variants.
  
 
=== Ultrafilter definition ===
 
=== Ultrafilter definition ===
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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.
 
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.
 
=== Ultrahuge cardinals, $C^{(n)}$-ultrahuge cardinals ===
 
 
''To be expanded.''
 
  
 
== Consistency strength and size ==
 
== Consistency strength and size ==
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* $n$-huge
 
* $n$-huge
 
* super $n$-huge
 
* super $n$-huge
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* ultra $n$-huge
 
* $(n+1)$-superstrong
 
* $(n+1)$-superstrong
  
 
All huge variants lay at the top of the double helix restricted to some [[Omega|natural number]] $n$, although each are bested by [[rank-into-rank|I3]] cardinals (the [[elementary embedding|critical points]] of the I3 elementary embeddings). In fact, every I3 is preceeded by a stationary set of $n$-huge cardinals, for all $n$. <cite>Kanamori2009:HigherInfinite</cite>
 
All huge variants lay at the top of the double helix restricted to some [[Omega|natural number]] $n$, although each are bested by [[rank-into-rank|I3]] cardinals (the [[elementary embedding|critical points]] of the I3 elementary embeddings). In fact, every I3 is preceeded by a stationary set of $n$-huge cardinals, for all $n$. <cite>Kanamori2009:HigherInfinite</cite>
  
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". <cite>Kanamori2009:HigherInfinite</cite>
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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" <cite>Kanamori2009:HigherInfinite</cite>, in fact it contains every cardinal $\lambda$ such that $V_\kappa\models$"$\lambda$ is ultra $n$-huge.
  
In terms of size, however, the least $n$-huge cardinal is smaller than the least [[supercompact]] cardinal (assuming both exist). <cite>Kanamori2009:HigherInfinite</cite> This is because $n$-huge cardinals have upward reflection properties, while supercompacts have downward reflection properties. Thus for any $\kappa$ which is supercompact and has an $n$-huge cardinal above it, $\kappa$ "reflects downward" that $n$-huge cardinal: there are $\kappa$-many $n$-huge cardinals below $\kappa$. On the other hand, the least super $n$-huge cardinals have ''both'' upward and downward reflection properties, and are all ''much'' larger than the least supercompact cardinal. It is notable that, while almost 2-huge cardinals have higher consistency strength than superhuge cardinals, the least almost 2-huge is much smaller than the least super almost huge.
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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 [[Vopenka|Vopěnka]] (therefore the consistency of the existence of an almost-huge cardinal implies the consistency of Vopěnka's principle). <cite>Kanamori2009:HigherInfinite</cite> Every ultra $n$-huge is super $n$-huge and a stationary limit of super $n$-huge cardinals. Every super almost $(n+1)$-huge is ultra $n$-huge and a stationary limit of ultra $n$-huge cardinals.
  
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 [[Vopenka|Vopěnka]] (therefore the consistency of the existence of an almost-huge cardinal implies the consistency of Vopěnka's principle). <cite>Kanamori2009:HigherInfinite</cite>
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In terms of size, however, the least $n$-huge cardinal is smaller than the least [[supercompact]] cardinal (assuming both exist). <cite>Kanamori2009:HigherInfinite</cite> This is because $n$-huge cardinals have upward reflection properties, while supercompacts have downward reflection properties. Thus for any $\kappa$ which is supercompact and has an $n$-huge cardinal above it, $\kappa$ "reflects downward" that $n$-huge cardinal: there are $\kappa$-many $n$-huge cardinals below $\kappa$. On the other hand, the least super $n$-huge cardinals have ''both'' upward and downward reflection properties, and are all ''much'' larger than the least supercompact cardinal. It is notable that, while almost 2-huge cardinals have higher consistency strength than superhuge cardinals, the least almost 2-huge is much smaller than the least super almost huge.
  
 
{{References}}
 
{{References}}

Revision as of 13:05, 8 December 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 $\sigma$-complete ideal on $\omega_1$". It is now known that only a Woodin cardinal is needed for this result. [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 a nontrivial elementary embedding of $V$ into a transitive class $M$ with critical point $\kappa$. 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, $M^{<\lambda}\subseteq M$).
  • $\kappa$ is $n$-huge with target $\lambda$ iff $\lambda=j^n(\kappa)$ and $M$ is closed under all of its sequences of length $\lambda$ ($M^\lambda\subseteq M$).
  • $\kappa$ 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.

Ultrahuge cardinals, $C^{(n)}$-ultrahuge cardinals

A cardinal $\kappa$ is $\lambda$-ultrahuge for $\lambda>\kappa$ if there exists a nontrivial elementary embedding $j:V\to M$ for some transitive class $M$ such that $\mathrm{j}(\kappa)>\lambda$, $M^{j(\kappa)}\subseteq M$ and $V_{j(\lambda)}\subseteq M$. A cardinal is ultrahuge if it is $\lambda$-ultrahuge for all $\lambda\geq\kappa$. [1] Notice how similar this definition is to the alternative characterization of extendible cardinals. Furthermore, this definition can be extended in the obvious way to define $\lambda$-ultra $n$-hugeness and ultra $n$-hugeness, as well as the "almost" variants.

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:

$$\forall i<n(\{x\subseteq\lambda:\text{order-type}(x\cap\lambda_{i+1})=\lambda_i\}\in U)$$

Where $\text{order-type}(X)$ 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 = 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
  • ultra $n$-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 fact it contains every cardinal $\lambda$ such that $V_\kappa\models$"$\lambda$ is ultra $n$-huge.

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] Every ultra $n$-huge is super $n$-huge and a stationary limit of super $n$-huge cardinals. Every super almost $(n+1)$-huge is ultra $n$-huge and a stationary limit of ultra $n$-huge cardinals.

In terms of size, however, the least $n$-huge cardinal is smaller than the least supercompact cardinal (assuming both exist). [1] This is because $n$-huge cardinals have upward reflection properties, while supercompacts have downward reflection properties. Thus for any $\kappa$ which is supercompact and has an $n$-huge cardinal above it, $\kappa$ "reflects downward" that $n$-huge cardinal: there are $\kappa$-many $n$-huge cardinals below $\kappa$. On the other hand, the least super $n$-huge cardinals have both upward and downward reflection properties, and are all much larger than the least supercompact cardinal. It is notable that, while almost 2-huge cardinals have higher consistency strength than superhuge cardinals, the least almost 2-huge is much smaller than the least super almost huge.

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 ofelementary embeddings. , 2007. www   bibtex
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