Difference between revisions of "Nearly supercompact"

From Cantor's Attic
Jump to: navigation, search
 
Line 1: Line 1:
 
{{DISPLAYTITLE: Nearly $\theta$-supercompact cardinals}}
 
{{DISPLAYTITLE: Nearly $\theta$-supercompact cardinals}}
  
The near $\theta$-supercompactness hierarchy of cardinals was introduced by Jason Schanker in <CITE>Schanker:PartialNearSupercompactness</CITE> and <CITE>Schanker2011:Thesis</CITE>.  The hierarchy stratifies the $\theta$-supercompactness hierarchy in the sense that every $\theta$-supercompact cardinal is nearly $\theta$-supercompact, and every nearly $2^{\theta^{{<}\kappa}}$-supercompact cardinal $\kappa$ is $\theta$-supercompact.  However, these cardinals can be very different.  For example, relative to the existence of a supercompact cardinal $\kappa$ with an inaccessible cardinal $\theta$ above it, we can force to destroy $\kappa$'s measurability while still retaining its near $\theta$-supercompactness and the [[weakly inaccessible|weak inaccessibility]] of $\theta$.  Yet, if $\theta^{{<}\kappa} = \theta$ and $\kappa$ is $\theta$-supercompact, we can also force to preserve $\kappa$'s $\theta$-supercompactness while destroying any potential near $\theta^+$-supercompactness without collapsing cardinals below $\theta^{++}$.  Assuming that $\theta^{{<}\kappa} = \theta$, nearly $\theta$-supercompact cardinals $\kappa$ exhibit a hybrid of [[weakly compact|weak compactness]] and [[supercompact|supercompactness]] in that the witnessing [[elementary embedding|embeddings]] are between $\text{ZFC}^-$ ($\text{ZFC}$ minus the powerset axiom) models of size $\theta$ but are generated by "partially normal" fine filters on $P_{\kappa}\theta$.  [[Weakly compact]] cardinals $\kappa$ are  nearly $\kappa$-supercompact.
+
The near $\theta$-supercompactness hierarchy of cardinals was introduced by Jason Schanker in <CITE>Schanker:PartialNearSupercompactness</CITE> and <CITE>Schanker2011:Thesis</CITE>.  The hierarchy stratifies the $\theta$-supercompactness hierarchy in the sense that every $\theta$-supercompact cardinal is nearly $\theta$-supercompact, and every nearly $2^{\theta^{{<}\kappa}}$-supercompact cardinal $\kappa$ is $\theta$-supercompact.  However, these cardinals can be very different.  For example, relative to the existence of a supercompact cardinal $\kappa$ with an inaccessible cardinal $\theta$ above it, we can force to destroy $\kappa$'s measurability while still retaining its near $\theta$-supercompactness and the [[weakly inaccessible|weak inaccessibility]] of $\theta$.  Yet, if $\theta^{{<}\kappa} = \theta$ and $\kappa$ is $\theta$-supercompact, we can also force to preserve $\kappa$'s $\theta$-supercompactness while destroying any potential near $\theta^+$-supercompactness without collapsing cardinals below $\theta^{++}$.  Assuming that $\theta^{{<}\kappa} = \theta$, nearly $\theta$-supercompact cardinals $\kappa$ exhibit a hybrid of [[weakly compact|weak compactness]] and [[supercompact|supercompactness]] in that the witnessing [[elementary embedding|embeddings]] are between $\text{ZFC}^-$ ($\text{ZFC}$ minus the powerset axiom) models of size $\theta$ but are generated by "partially normal" fine [[filter|filters]] on $P_{\kappa}\theta$.  [[Weakly compact]] cardinals $\kappa$ are  nearly $\kappa$-supercompact.
 
   
 
   
 
== Formal definition ==
 
== Formal definition ==
Line 17: Line 17:
 
:; Normal ZFC Embedding :
 
:; Normal ZFC Embedding :
  
:; Normal Fine Filter : For every family of subsets $\mathcal{A} \subset P_\kappa\theta$ of size at most $\theta$ and every collection $\mathcal{F}$ of at most $\theta$ many functions from $P_{\kappa}\theta$ into $\theta$, there exists a $\kappa$-complete fine filter $F$ on $P_{\kappa}\theta$, which is $\mathcal{F}$-normal in the sense that for every $f \in \mathcal{F}$ that's regressive on some set in $F$, there exists $\alpha_f < \theta$ for which $\{\sigma \in P_{\kappa}\theta| f(\sigma) = \alpha_f\} \in F$.
+
:; Normal Fine Filter : For every family of subsets $\mathcal{A} \subset P_\kappa\theta$ of size at most $\theta$ and every collection $\mathcal{F}$ of at most $\theta$ many functions from $P_{\kappa}\theta$ into $\theta$, there exists a $\kappa$-complete fine [[filter]] $F$ on $P_{\kappa}\theta$, which is $\mathcal{F}$-normal in the sense that for every $f \in \mathcal{F}$ that's regressive on some set in $F$, there exists $\alpha_f < \theta$ for which $\{\sigma \in P_{\kappa}\theta| f(\sigma) = \alpha_f\} \in F$.
  
 
:; Hauser Embedding :
 
:; Hauser Embedding :

Latest revision as of 01:16, 10 October 2017


The near $\theta$-supercompactness hierarchy of cardinals was introduced by Jason Schanker in [1] and [2]. The hierarchy stratifies the $\theta$-supercompactness hierarchy in the sense that every $\theta$-supercompact cardinal is nearly $\theta$-supercompact, and every nearly $2^{\theta^{{<}\kappa}}$-supercompact cardinal $\kappa$ is $\theta$-supercompact. However, these cardinals can be very different. For example, relative to the existence of a supercompact cardinal $\kappa$ with an inaccessible cardinal $\theta$ above it, we can force to destroy $\kappa$'s measurability while still retaining its near $\theta$-supercompactness and the weak inaccessibility of $\theta$. Yet, if $\theta^{{<}\kappa} = \theta$ and $\kappa$ is $\theta$-supercompact, we can also force to preserve $\kappa$'s $\theta$-supercompactness while destroying any potential near $\theta^+$-supercompactness without collapsing cardinals below $\theta^{++}$. Assuming that $\theta^{{<}\kappa} = \theta$, nearly $\theta$-supercompact cardinals $\kappa$ exhibit a hybrid of weak compactness and supercompactness in that the witnessing embeddings are between $\text{ZFC}^-$ ($\text{ZFC}$ minus the powerset axiom) models of size $\theta$ but are generated by "partially normal" fine filters on $P_{\kappa}\theta$. Weakly compact cardinals $\kappa$ are nearly $\kappa$-supercompact.

Formal definition

A cardinal $\kappa$ is nearly $\theta$-supercompact if and only if for every $A \subseteq \theta$, there exists a transitive $M \vDash ZFC^{-}$ closed under ${<}\kappa$ sequences with $A, \kappa, \theta \in M$, a transitive $N$, and an elementary embedding $j: M \rightarrow N$ with critical point $\kappa$ such that $j(\kappa) > \theta$ and $j''\theta \in N$. A cardinal is nearly supercompact if it is nearly $\theta$-supercompact for all $\theta$.

Characterizations of near $\theta$-supercompactness

If $\theta^{{<}\kappa} = \theta$, then the following are equivalent characterizations for the near $\theta$-supercompactness of $\kappa$:

Embedding 
For every ${<}\kappa$-closed transitive set $M$ of size $\theta$ with $\theta \in M$, there exists a transitive $N$ and an elementary embedding $j: M \rightarrow N$ with critical point $\kappa$ such that $j(\kappa) > \theta$ and $j''\theta \in N$.
Normal Embedding 
Normal ZFC Embedding 
Normal Fine Filter 
For every family of subsets $\mathcal{A} \subset P_\kappa\theta$ of size at most $\theta$ and every collection $\mathcal{F}$ of at most $\theta$ many functions from $P_{\kappa}\theta$ into $\theta$, there exists a $\kappa$-complete fine filter $F$ on $P_{\kappa}\theta$, which is $\mathcal{F}$-normal in the sense that for every $f \in \mathcal{F}$ that's regressive on some set in $F$, there exists $\alpha_f < \theta$ for which $\{\sigma \in P_{\kappa}\theta| f(\sigma) = \alpha_f\} \in F$.
Hauser Embedding 

Nearly strongly compact

References

  1. Schanker, Jason A. Partial near supercompactness. Ann Pure Appl Logic , 2012. (In Press.) www   DOI   bibtex
  2. Schanker, Jason A. Weakly measurable cardinals and partial near supercompactness. Ph.D. Thesis, CUNY Graduate Center, 2011. bibtex
Main library