Difference between revisions of "Unfoldable"

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=== Unfoldable ===
 
=== Unfoldable ===
  
A cardinal $\kappa$ is '''$\theta$-unfoldable''' for every transitive set $M\models\text{ZFC}$ such that $|M|=\kappa$ and $\kappa\in M$, there is some transitive $N$ and $j:M\rightarrow N$ and elementary embedding with critical point $\kappa$ and $j(\kappa)\geq\theta$. $\kappa$ is then called '''unfoldable''' iff it is $\theta$-unfoldable for every $\theta$; i.e. the target of the embedding can be made arbitrarily large.
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A cardinal $\kappa$ is '''$\theta$-unfoldable''' iff for every $A\subseteq\kappa$, there is some transitive $M$ with $A\in M\models\text{ZFC}$ and some $j:M\rightarrow N$ an elementary embedding with critical point $\kappa$ such that $j(\kappa)\geq\theta$. $\kappa$ is then called '''unfoldable''' iff it is $\theta$-unfoldable for every $\theta$; i.e. the target of the embedding can be made arbitrarily large.
  
 
Unfoldability can also be characterized by the order-type of nontrivial elementary end-extensions of $V_\kappa$. Let $\mathcal{M}=(M,R_0^\mathcal{M},R_1^\mathcal{M}...)$ be an aribtrary structure of type $(\alpha,\beta)$ with relations $R_0^\mathcal{M},R_1^\mathcal{M}...$ and $\mathcal{N}=(N,R_0^\mathcal{N},R_1^\mathcal{N}...)$ be another arbitrary structure of the same language with relations $R_0^\mathcal{N},R_1^\mathcal{N}...$. Then, one writes $\mathcal{M}\prec_e\mathcal{N}$ iff all of the following hold:
 
Unfoldability can also be characterized by the order-type of nontrivial elementary end-extensions of $V_\kappa$. Let $\mathcal{M}=(M,R_0^\mathcal{M},R_1^\mathcal{M}...)$ be an aribtrary structure of type $(\alpha,\beta)$ with relations $R_0^\mathcal{M},R_1^\mathcal{M}...$ and $\mathcal{N}=(N,R_0^\mathcal{N},R_1^\mathcal{N}...)$ be another arbitrary structure of the same language with relations $R_0^\mathcal{N},R_1^\mathcal{N}...$. Then, one writes $\mathcal{M}\prec_e\mathcal{N}$ iff all of the following hold:
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A cardinal $\kappa$ is '''unfoldable''' iff $\kappa$ is [[inaccessible]] and for any $S\subset V_\kappa$, there are well-founded models $M$ nontrivially end elementary extending $(V_\kappa;\in,S)$ of arbitrarily large rank. In this case, $(V_\kappa;\in,S)\prec_e (M;\in^M,S^M)$ iff $(V_\kappa;\in,S)\prec (M;\in^M,S^M)$ and $(V_\kappa;\in)\prec_e (M;\in^M)$. <cite>Villaveces1996:ChainsEndElementaryExtensionsModels</cite>
 
A cardinal $\kappa$ is '''unfoldable''' iff $\kappa$ is [[inaccessible]] and for any $S\subset V_\kappa$, there are well-founded models $M$ nontrivially end elementary extending $(V_\kappa;\in,S)$ of arbitrarily large rank. In this case, $(V_\kappa;\in,S)\prec_e (M;\in^M,S^M)$ iff $(V_\kappa;\in,S)\prec (M;\in^M,S^M)$ and $(V_\kappa;\in)\prec_e (M;\in^M)$. <cite>Villaveces1996:ChainsEndElementaryExtensionsModels</cite>
  
$\kappa$ is also unfoldable iff for any $S\subseteq\kappa$, letting $\mathcal{E}$ be the class of all eees of $(V_\kappa;\in,S)$, $(\mathcal{E};\prec_e)$ has abitrarily long chains. The name "unfoldable" comes from the fact that "unfolding" $(V_\kappa;\in,S)$ yields a larger structure with the same properties and a bit of excess information, and this can be done arbitrarily many times on the iterated results of "unfolding". <cite>Villaveces1996:ChainsEndElementaryExtensionsModels</cite>
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$\kappa$ is also '''unfoldable''' iff for any $S\subseteq\kappa$, letting $\mathcal{E}$ be the class of all eees of $(V_\kappa;\in,S)$, $(\mathcal{E};\prec_e)$ has abitrarily long chains. The name "unfoldable" comes from the fact that "unfolding" $(V_\kappa;\in,S)$ yields a larger structure with the same properties and a bit of excess information, and this can be done arbitrarily many times on the iterated results of "unfolding". <cite>Villaveces1996:ChainsEndElementaryExtensionsModels</cite>
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=== Long Unfoldable ===
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$\kappa$ is '''long unfoldable''' iff for any $S\subseteq\kappa$, letting $\mathcal{E}$ be the class of all eees of $(V_\kappa;\in,S)$, $(\mathcal{E};\prec_e)$ has chains of length $\text{Ord}$.
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The least unfoldable cardinal is never long unfoldable. Every long unfoldable cardinal is unfoldable. <cite>Villaveces1996:ChainsEndElementaryExtensionsModels</cite>
  
 
=== Strongly Unfoldable ===
 
=== Strongly Unfoldable ===
  
A cardinal $\kappa$ is '''$\theta$-strongly unfoldable''' iff for every transitive set $M\models\text{ZFC}$ such that $|M|=\kappa$ and $\kappa\in M$, there is some transitive $N\supseteq V_\theta$ and $j:M\rightarrow N$ and elementary embedding with critical point $\kappa$ and $j(\kappa)\geq\theta$.
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A cardinal $\kappa$ is '''$\theta$-strongly unfoldable''' iff for every $A\subseteq\kappa$, there is some transitive $M$ with $A\in M\models\text{ZFC}$ and some $j:M\rightarrow N$ an elementary embedding with critical point $\kappa$ such that $j(\kappa)\geq\theta$ and $V_\theta\subseteq N$.
  
 
$\kappa$ is then called '''strongly unfoldable''' iff it is $\theta$-strongly unfoldable for every $\theta$; i.e. the target of the embedding can be made arbitrarily large.
 
$\kappa$ is then called '''strongly unfoldable''' iff it is $\theta$-strongly unfoldable for every $\theta$; i.e. the target of the embedding can be made arbitrarily large.
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=== Superstrongly Unfoldable ===
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A cardinal $\kappa$ is '''$\theta$-superstrongly unfoldable''' iff for every $A\subseteq\kappa$, there is some transitive $M$ with $A\in M\models\text{ZFC}$ and some $j:M\rightarrow N$ an elementary embedding with critical point $\kappa$ such that $j(\kappa)\geq\theta$ and $V_{j(\kappa)}\subseteq N$.
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$\kappa$ is then called '''superstrongly unfoldable''' iff it is $\theta$-strongly unfoldable for every $\theta$; i.e. the target of the embedding can be made arbitrarily large. Equivalently, $\kappa$ is [[uplifting|strongly uplifting]].
  
 
== Relations to Other Cardinals ==
 
== Relations to Other Cardinals ==
 
Here is a list of relations between unfoldability and other large cardinal axioms:
 
Here is a list of relations between unfoldability and other large cardinal axioms:
  
*The $\kappa$ which are $\kappa$-unfoldable are precisely those which are $\kappa$-strongly unfoldable. Furthermore, if $V=L$, then any $\theta$-unfoldable cardinal is $\theta$-strongly unfoldable. Therefore, unfoldability and strong unfoldability are equiconsistent. <cite>Hamkins2008:UnfoldableGCH</cite>
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*The $\kappa$ which are $\kappa$-unfoldable are precisely those which are $\kappa$-strongly unfoldable, which are precisely those that are [[weakly compact]]. Furthermore, if $V=L$, then any $\theta$-unfoldable cardinal is $\theta$-strongly unfoldable. Therefore, unfoldability and strong unfoldability are equiconsistent. <cite>Hamkins2008:UnfoldableGCH</cite>
 
*The assertion that a [[Ramsey]] cardinal and a strongly unfoldable cardinal both exists is stronger than the assertion that there exists both an unfoldable cardinal and a strongly unfoldable cardinal. <cite>Villaveces1996:ChainsEndElementaryExtensionsModels</cite>
 
*The assertion that a [[Ramsey]] cardinal and a strongly unfoldable cardinal both exists is stronger than the assertion that there exists both an unfoldable cardinal and a strongly unfoldable cardinal. <cite>Villaveces1996:ChainsEndElementaryExtensionsModels</cite>
 
*Interestingly, if there is an unfoldable cardinal then there is a forcing extension in which all unfoldable cardinals in $V$ are unfoldable in the forcing extension and GCH fails at every [[inaccessible]] cardinal. Therefore, GCH can fail at every unfoldable cardinal. <cite>Hamkins2008:UnfoldableGCH</cite>
 
*Interestingly, if there is an unfoldable cardinal then there is a forcing extension in which all unfoldable cardinals in $V$ are unfoldable in the forcing extension and GCH fails at every [[inaccessible]] cardinal. Therefore, GCH can fail at every unfoldable cardinal. <cite>Hamkins2008:UnfoldableGCH</cite>

Revision as of 11:52, 5 December 2017

The unfoldable cardinals were introduced by Andres Villaveces in order to generalize the definition of weak compactness. Because weak compactness has many different definitions, the one he chose to extend was specifically the embedding property (see weakly compact for more information). The way he did this was analogous to the generalization of huge cardinals to superhuge cardinals.

Definition

There are unfoldable cardinals and strongly unfoldable cardinals, as well as superstrongly unfoldable cardinals. All of these are generalizations of weak compactness.

Unfoldable

A cardinal $\kappa$ is $\theta$-unfoldable iff for every $A\subseteq\kappa$, there is some transitive $M$ with $A\in M\models\text{ZFC}$ and some $j:M\rightarrow N$ an elementary embedding with critical point $\kappa$ such that $j(\kappa)\geq\theta$. $\kappa$ is then called unfoldable iff it is $\theta$-unfoldable for every $\theta$; i.e. the target of the embedding can be made arbitrarily large.

Unfoldability can also be characterized by the order-type of nontrivial elementary end-extensions of $V_\kappa$. Let $\mathcal{M}=(M,R_0^\mathcal{M},R_1^\mathcal{M}...)$ be an aribtrary structure of type $(\alpha,\beta)$ with relations $R_0^\mathcal{M},R_1^\mathcal{M}...$ and $\mathcal{N}=(N,R_0^\mathcal{N},R_1^\mathcal{N}...)$ be another arbitrary structure of the same language with relations $R_0^\mathcal{N},R_1^\mathcal{N}...$. Then, one writes $\mathcal{M}\prec_e\mathcal{N}$ iff all of the following hold:

  • $\mathcal{M}$ is an elementary substructure of $\mathcal{N}$
  • $\mathcal{M}\neq\mathcal{N}$
  • For any $a\in M$, $b\in N$, and $\gamma<\beta$, $b R_\gamma^\mathcal{N} a\rightarrow b\in M$

If such holds, $\mathcal{M}$ is said to be nontrivially end elementary extended by $\mathcal{N}$ and $\mathcal{N}$ is a nontrivial end elementary extension of $\mathcal{M}$, abbreviated $\mathcal{N}$ is an eee of $\mathcal{M}$.

A cardinal $\kappa$ is unfoldable iff $\kappa$ is inaccessible and for any $S\subset V_\kappa$, there are well-founded models $M$ nontrivially end elementary extending $(V_\kappa;\in,S)$ of arbitrarily large rank. In this case, $(V_\kappa;\in,S)\prec_e (M;\in^M,S^M)$ iff $(V_\kappa;\in,S)\prec (M;\in^M,S^M)$ and $(V_\kappa;\in)\prec_e (M;\in^M)$. [1]

$\kappa$ is also unfoldable iff for any $S\subseteq\kappa$, letting $\mathcal{E}$ be the class of all eees of $(V_\kappa;\in,S)$, $(\mathcal{E};\prec_e)$ has abitrarily long chains. The name "unfoldable" comes from the fact that "unfolding" $(V_\kappa;\in,S)$ yields a larger structure with the same properties and a bit of excess information, and this can be done arbitrarily many times on the iterated results of "unfolding". [1]

Long Unfoldable

$\kappa$ is long unfoldable iff for any $S\subseteq\kappa$, letting $\mathcal{E}$ be the class of all eees of $(V_\kappa;\in,S)$, $(\mathcal{E};\prec_e)$ has chains of length $\text{Ord}$.

The least unfoldable cardinal is never long unfoldable. Every long unfoldable cardinal is unfoldable. [1]

Strongly Unfoldable

A cardinal $\kappa$ is $\theta$-strongly unfoldable iff for every $A\subseteq\kappa$, there is some transitive $M$ with $A\in M\models\text{ZFC}$ and some $j:M\rightarrow N$ an elementary embedding with critical point $\kappa$ such that $j(\kappa)\geq\theta$ and $V_\theta\subseteq N$.

$\kappa$ is then called strongly unfoldable iff it is $\theta$-strongly unfoldable for every $\theta$; i.e. the target of the embedding can be made arbitrarily large.

Superstrongly Unfoldable

A cardinal $\kappa$ is $\theta$-superstrongly unfoldable iff for every $A\subseteq\kappa$, there is some transitive $M$ with $A\in M\models\text{ZFC}$ and some $j:M\rightarrow N$ an elementary embedding with critical point $\kappa$ such that $j(\kappa)\geq\theta$ and $V_{j(\kappa)}\subseteq N$.

$\kappa$ is then called superstrongly unfoldable iff it is $\theta$-strongly unfoldable for every $\theta$; i.e. the target of the embedding can be made arbitrarily large. Equivalently, $\kappa$ is strongly uplifting.

Relations to Other Cardinals

Here is a list of relations between unfoldability and other large cardinal axioms:

  • The $\kappa$ which are $\kappa$-unfoldable are precisely those which are $\kappa$-strongly unfoldable, which are precisely those that are weakly compact. Furthermore, if $V=L$, then any $\theta$-unfoldable cardinal is $\theta$-strongly unfoldable. Therefore, unfoldability and strong unfoldability are equiconsistent. [2]
  • The assertion that a Ramsey cardinal and a strongly unfoldable cardinal both exists is stronger than the assertion that there exists both an unfoldable cardinal and a strongly unfoldable cardinal. [1]
  • Interestingly, if there is an unfoldable cardinal then there is a forcing extension in which all unfoldable cardinals in $V$ are unfoldable in the forcing extension and GCH fails at every inaccessible cardinal. Therefore, GCH can fail at every unfoldable cardinal. [2]
  • Although unfoldable cardinals are consistency-wise stronger than weakly compact cardinals, if there is a strongly unfoldable cardinal, then in a forcing extension, the least weakly compact cardinal is also the least unfoldable cardinal.[3]
  • The existence of a subtle cardinal is consistency-wise stronger than the existence of an unfoldable cardinal. [1]
  • If a subtle cardinal and an unfoldable cardinal exist and $V=L$, then the least unfoldable cardinal is larger than the least subtle cardinal (and therefore much larger than the least weakly compact). [1]
  • Any Ramsey cardinal is unfoldable. [1]
  • Any strongly unfoldable cardinal is totally indescribable. [1]
  • Even though it may seem absurd, if both exist then the least I3 cardinal is less than the least strongly unfoldable cardinal.

Relation to forcing

e.g. GCH, indestructibility, connection to weak forms of PFA

consistency with slim Kurepa trees

References

  1. Villaveces, Andrés. Chains of End Elementary Extensions of Models of Set Theory. JSTOR , 1996. www   bibtex
  2. Hamkins, Joel David. Unfoldable cardinals and the GCH. , 2008. www   bibtex
  3. Cody, Brent, Gitik, Moti, Hamkins, Joel David, and Schanker, Jason. The Least Weakly Compact Cardinal Can Be Unfoldable, Weakly Measurable and Nearly θ-Supercompact. , 2013. www   bibtex
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