Difference between revisions of "Supercompact"

From Cantor's Attic
Jump to: navigation, search
Line 7: Line 7:
 
Generalizing the [[elementary embedding]] characterization of measurable cardinal, a cardinal $\kappa$ is ''$\theta$-supercompact'' if there is an elementary embedding $j:V\to M$ with $M$ a transitive class, such that $j$ has critical point $\kappa$ and $M^\theta\subset M$, i.e. $M$ is closed under arbitrary sequences of length $\theta$. Under the [[axiom of choice]], one may assume without loss of generality that $j(\kappa)\gt\theta$. $\kappa$ is then said to be ''supercompact'' if it is $\theta$-supercompact for all $\theta$. It is worth noting that, using this formulation, the collection of sets of [[Hereditary Cardinality]] less than $\theta^+$ must be contained in the transitive class $M$.  
 
Generalizing the [[elementary embedding]] characterization of measurable cardinal, a cardinal $\kappa$ is ''$\theta$-supercompact'' if there is an elementary embedding $j:V\to M$ with $M$ a transitive class, such that $j$ has critical point $\kappa$ and $M^\theta\subset M$, i.e. $M$ is closed under arbitrary sequences of length $\theta$. Under the [[axiom of choice]], one may assume without loss of generality that $j(\kappa)\gt\theta$. $\kappa$ is then said to be ''supercompact'' if it is $\theta$-supercompact for all $\theta$. It is worth noting that, using this formulation, the collection of sets of [[Hereditary Cardinality]] less than $\theta^+$ must be contained in the transitive class $M$.  
  
There is an alternative formulation that is expressible in ZFC using certain [[ultrafilter]]s with somewhat technical properties: For $\kappa <\theta$, recall that $P_\kappa\theta$ denotes the collection of subsets of $\theta$ of size smaller than $\kappa$. Call an ultrafilter $U\subseteq P_\kappa\theta$ ''fine'' if for every $\alpha < \theta$ the set $\{a\in P_\kappa\theta: \alpha\in a\}$ is a member of $U$. Call an ultrafilter $U\subseteq P_\kappa\theta$ ''normal'' if $U$ is $\theta$-complete and for every "regressive" function $f:P_\kappa\theta\to\theta$ with $\{a\in P_\kappa\theta: f(a)\in a\}\in U$ there is some $\alpha <\theta$ where $\{a\in P_\kappa\theta: f(a)=\alpha\}\in U$. (An equivalent notion of ''normal'' using diagonal intersections can be found in Kanamori.)
+
There is an alternative formulation that is expressible in ZFC using certain [[ultrafilter]]s with somewhat technical properties: for $\theta\geq\kappa$, $\kappa$ if $\theta$-supercompact if there is a normal fine measure on $P_\kappa(\theta)$. $\kappa$ is supercompact if for every set $A$ with $|A|\geq\kappa$, there is a normal fine measure on $P_\kappa(A)$.
  
Using these notions, one can define $\kappa$ as ''$\theta$-supercompact'' if there is a normal fine ultrafilter $U$ on $P_\kappa\theta$. $\kappa$ is ''supercomact'', then, if there is such an ultrafilter $U\subseteq P_\kappa\theta$ for every cardinal $\theta$ at least as big as $\kappa$.
+
One can see the equivalence of the two formulations by first considering the ultrafilter $U$ arising from the [[seed]] $j''\theta$, so that $X\in U\iff j''\theta\in j(X)$. It is easy to check that $U$ is a normal fine measure on $P_\kappa\theta$. Conversely, the ultrapower by a normal fine measure $U$ on $P_\kappa\theta$ gives rise to an embedding $j:V\to M$ (here $M$ is identified with the transitive collapse of the ultrapower by $U$). It is then straightforward to check that $\theta$ is the critical point of this embedding and that $M$ is sufficiently closed, thus witnessing $\theta$-supercompactness of $\kappa$.  
  
One can see the equivalence of the two formulations by first considering the ultrafilter $U$ arising from the [[seed]] $j''\theta$, so that $X\in U\iff j''\theta\in j(X)$. It is easy to check that $U$ is a normal fine measure on $P_\kappa\theta$. Conversely, the ultrapower by a normal fine ultrafilter $U$ on $P_\kappa\theta$ gives rise to an embedding $j:V\to M$ (here $M$ is identified with the transitive collapse of the ultrapower by $U$). It is then straightforward to check that $\theta$ is the critical point of this embedding and that $M$ is sufficiently closed, thus witnessing $\theta$-supercompactness of $\kappa$.
+
A third characterization was given by Magidor in terms of elementary embeddings from initial segments of $V$ into other (larger) initial segments of $V$, but in this characterization, the supercompact cardinal $\kappa$ is the ''image'' of the critical point of this embedding, rather than the critical point itself.  
 
+
A third characterization was given by Magidor in terms of elementary embeddings from initial segments of $V$ into other (larger) initial segments of $V$, but in this characterization, the supercompact cardinal $\kappa$ is the ''image'' of the critical point of this embedding, rather than the critical point itself. See...  
+
  
 
== Properties ==
 
== Properties ==
  
If $\kappa$ is supercompact, then there are $2^{2^\kappa}$ normal fine [[ultrafilter|ultrafilters]] on $\kappa$, also for every $\lambda\geq\kappa$ there are $2^{2^{\lambda^{<\kappa}}}$ normal fine ultrafilters on $P_\kappa(\lambda)$.
+
If $\kappa$ is supercompact, then there are $2^{2^\kappa}$ [[filter|normal fine measures]] on $\kappa$, also for every $\lambda\geq\kappa$ there are $2^{2^{\lambda^{<\kappa}}}$ normal fine measures on $P_\kappa(\lambda)$.
  
 
Every supercompact has [[Mitchell order]] $(2^\kappa)^+\geq\kappa^{++}$.
 
Every supercompact has [[Mitchell order]] $(2^\kappa)^+\geq\kappa^{++}$.
Line 23: Line 21:
 
If $\lambda\geq\kappa$ is regular, $\kappa$ is $\lambda$-supercompact, then every $\alpha<\kappa$ that is $\gamma$-supercompact for $\gamma<\kappa$ (if any exists) is also $\lambda$-supercompact.
 
If $\lambda\geq\kappa$ is regular, $\kappa$ is $\lambda$-supercompact, then every $\alpha<\kappa$ that is $\gamma$-supercompact for $\gamma<\kappa$ (if any exists) is also $\lambda$-supercompact.
  
''Laver's theorem'' asserts that if $\kappa$ is supercompact, there exists a function $f:\kappa\to V_\kappa$ such that for every $x$ and $\lambda\geq\kappa$ with $|tc(x)|\leq\lambda$ there exists a normal fine ultrafilter $U$ on $P_\kappa(\lambda)$ such that $j_U(f)(\kappa)=x$, where $j_U$ is the elementary embedding generated from $U$. Here $tc(x)$ is the ''transitive closure'' of $x$ (i.e. the smallest transitive set containing $x$), and $f$ is called a ''Laver function''.
+
''Laver's theorem'' asserts that if $\kappa$ is supercompact, there exists a function $f:\kappa\to V_\kappa$ such that for every $x$ and $\lambda\geq\kappa$ with $|tc(x)|\leq\lambda$ there exists a normal fine measure $U$ on $P_\kappa(\lambda)$ such that $j_U(f)(\kappa)=x$, where $j_U$ is the elementary embedding generated from $U$. Here $tc(x)$ is the ''transitive closure'' of $x$ (i.e. the smallest transitive set containing $x$), and $f$ is called a ''Laver function''.
  
 
===Reflection Properties===
 
===Reflection Properties===

Revision as of 01:31, 10 October 2017

Supercompact cardinals are best motivated as a generalization of measurable cardinals, particularly the characterization of measurable cardinals in terms of elementary embeddings and strong closure properties. The notion of supercompactness and its consequences was initially developed by Solovay and Reinhardt and further elaborated on by Magidor and Gitik, among many others. Assuming the existence of a supercompact is a very strong assumption and the large cardinal strength of supercompact cardinals is seen in a wide (and bewildering) array of set-theoretic contexts, especially the development of strong forcing axioms and establishing regularity properties of sets of reals. The inner model program has yet to reach the level of a supercompact cardinal and this is considered a prominent open problem in the program itself. Curiously, by results of Woodin, should the inner program reach the level of a supercompact, there is a sense in which it will have reached all greater large cardinals, a startling contrast to previous advances in the program.


Formal definition and equivalent characterizations

Generalizing the elementary embedding characterization of measurable cardinal, a cardinal $\kappa$ is $\theta$-supercompact if there is an elementary embedding $j:V\to M$ with $M$ a transitive class, such that $j$ has critical point $\kappa$ and $M^\theta\subset M$, i.e. $M$ is closed under arbitrary sequences of length $\theta$. Under the axiom of choice, one may assume without loss of generality that $j(\kappa)\gt\theta$. $\kappa$ is then said to be supercompact if it is $\theta$-supercompact for all $\theta$. It is worth noting that, using this formulation, the collection of sets of Hereditary Cardinality less than $\theta^+$ must be contained in the transitive class $M$.

There is an alternative formulation that is expressible in ZFC using certain ultrafilters with somewhat technical properties: for $\theta\geq\kappa$, $\kappa$ if $\theta$-supercompact if there is a normal fine measure on $P_\kappa(\theta)$. $\kappa$ is supercompact if for every set $A$ with $|A|\geq\kappa$, there is a normal fine measure on $P_\kappa(A)$.

One can see the equivalence of the two formulations by first considering the ultrafilter $U$ arising from the seed $j''\theta$, so that $X\in U\iff j''\theta\in j(X)$. It is easy to check that $U$ is a normal fine measure on $P_\kappa\theta$. Conversely, the ultrapower by a normal fine measure $U$ on $P_\kappa\theta$ gives rise to an embedding $j:V\to M$ (here $M$ is identified with the transitive collapse of the ultrapower by $U$). It is then straightforward to check that $\theta$ is the critical point of this embedding and that $M$ is sufficiently closed, thus witnessing $\theta$-supercompactness of $\kappa$.

A third characterization was given by Magidor in terms of elementary embeddings from initial segments of $V$ into other (larger) initial segments of $V$, but in this characterization, the supercompact cardinal $\kappa$ is the image of the critical point of this embedding, rather than the critical point itself.

Properties

If $\kappa$ is supercompact, then there are $2^{2^\kappa}$ normal fine measures on $\kappa$, also for every $\lambda\geq\kappa$ there are $2^{2^{\lambda^{<\kappa}}}$ normal fine measures on $P_\kappa(\lambda)$.

Every supercompact has Mitchell order $(2^\kappa)^+\geq\kappa^{++}$.

If $\lambda\geq\kappa$ is regular, $\kappa$ is $\lambda$-supercompact, then every $\alpha<\kappa$ that is $\gamma$-supercompact for $\gamma<\kappa$ (if any exists) is also $\lambda$-supercompact.

Laver's theorem asserts that if $\kappa$ is supercompact, there exists a function $f:\kappa\to V_\kappa$ such that for every $x$ and $\lambda\geq\kappa$ with $|tc(x)|\leq\lambda$ there exists a normal fine measure $U$ on $P_\kappa(\lambda)$ such that $j_U(f)(\kappa)=x$, where $j_U$ is the elementary embedding generated from $U$. Here $tc(x)$ is the transitive closure of $x$ (i.e. the smallest transitive set containing $x$), and $f$ is called a Laver function.

Reflection Properties

Supercompact cardinals and forcing

The continuum hypothesis and supercompact cardinals

If $\kappa$ is $\lambda$-supercompact and $2^\alpha=\alpha^{+}$ for every $\alpha<\kappa$, then $2^\alpha=\alpha^{+}$ for every $\alpha\leq\lambda$. Consequently, if the generalized continuum hypothesis holds below a supercompact cardinal, then it holds everywhere.

The existence of a supercompact implies the consistency of the failure of the singular cardinal hypothesis, i.e. it is consistent that the generalized continuum hypothesis fails at a strong limit singular cardinal. It also implies the consistency of the failure of the $GCH$ at a measurable cardinal.

By combining results of Magidor, Shelah and Gitik, one can show that the existence of a supercompact also implies the existence of a generic extension in which $2^{\aleph_\alpha}<\aleph_{\omega_1}$ for all $\alpha<\omega_1$, but also $2^{\aleph_{\omega_1}}>\aleph_{\omega_1+\alpha+1}$ for any prescribed $\alpha<\omega_2$. Similarly, one can have a generic extension in which the $GCH$ holds below $\aleph_\omega$ but $2^{\aleph_\omega}>\aleph_{\omega+\alpha+1}$ for any prescribed $\alpha<\omega_1$.

Woodin and Cummings furthermore showed that if there exists a supercompact, then there is a generic extension in which $2^\kappa=\kappa^{++}$ for every cardinal $\kappa$, i.e. the $GCH$ fails everywhere(!).

Laver preparation

Indestructibility, including the Laver diamond.

Proper forcing axiom

Baumgartner proved that if there is a supercompact cardinal, then the proper forcing axiom holds in a forcing extenion. $PFA$'s strengthening, $PFA^{+}$, is also consistent relative to the existence of a supercompact cardinal.

Martin's Maximum

Relation to other large cardinals

If a cardinal $\theta$-supercompact then it also $\theta$-strongly compact. Consequently, every supercompact cardinal is also strongly compact. It is consistent with $ZFC$ that every strongly compact cardinal is also supercompact, but it is not currently known whether the existence of a strongly compact cardinal is equiconsistent with the existence of a supercompact cardinal.

The least supercompact is larger than the least huge cardinal (if such a cardinal exists).

If there exists a measurable cardinal that is a limit of strongly compact cardinals, then the least such cardinal is strongly compact but not supercompact.


    This article is a stub. Please help us to improve Cantor's Attic by adding information.