Difference between revisions of "Nearly supercompact"
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{{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 == | ||
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:; 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 : |
Revision as of 00:16, 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.
Contents
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
- Schanker, Jason A. Partial near supercompactness. Ann Pure Appl Logic , 2012. (In Press.) www DOI bibtex
- Schanker, Jason A. Weakly measurable cardinals and partial near supercompactness. Ph.D. Thesis, CUNY Graduate Center, 2011. bibtex