Difference between revisions of "Jonsson"

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{{DISPLAYTITLE: Jónsson cardinal}}
 
{{DISPLAYTITLE: Jónsson cardinal}}
Jónsson cardinals are named after Bjarni Jónsson, a student of Tarski working in universal algebra. In 1962 he asked whether or or not every algebra of cardinality $\kappa$ have a subalgebra of the same cardinality. The cardinals $\kappa$ that satisfy this property are now called '''Jónsson cardinals'''. Equivalent definitions of Jónsson cardinals include the following - see 8.12 in Kanamori (2009) and Tryba (1984). A cardinal $\kappa$ is Jónsson iff:
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[[Category:Large cardinal axioms]]
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[[Category:Partition property]]
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Jónsson cardinals are named after Bjarni Jónsson, a student of Tarski working in universal algebra. In 1962, he asked whether or or not every algebra of cardinality $\kappa$ has a proper subalgebra of the same cardinality. The cardinals $\kappa$ that satisfy this property are now called '''Jónsson cardinals'''.
  
'''Combinatorial definition'''
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An algebra of cardinality $\kappa$ is simply a set $A$ of cardinality $\kappa$ and finitely many functions (each with finitely many inputs) $f_0,f_1...f_n$ for which $A$ is closed under every such function. A subalgebra of such an algebra is a set $B\subseteq A$ such that $B$ is closed under
* $\kappa\rightarrow [\kappa]_\kappa^{<\omega}$, i.e. that given any function $f:[\kappa]^{<\omega}\to\kappa$ we can find a subset $H\subseteq\kappa$ such that $f"[H]^n\neq\kappa$ for every $n<\omega$;
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'''Model-theoretic definitions'''
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== Equivalent Definitions ==
*  Any first-order structure with universe of cardinality $\kappa$ has a proper elementary substructure with universe also having cardinality $\kappa$;
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* Given any $A$ there exists an elementary substructure $\langle X,\in, X\cap A\rangle\prec\langle V_\kappa,\in,A\rangle$ with $|X|=\kappa$ and $X\cap\kappa\neq\kappa$;
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'''Embedding definitions'''
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There are several equivalent definitions to Jónsson cardinals cardinals.
* For every $\theta>\kappa$ there exists a transitive set $M$ with $\kappa\in M$ and an elementary embedding $j:M\to H_\theta$ such that $j(\kappa)=\kappa$ and $\text{crit }j<\kappa$;
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* There exists a $\theta>\kappa$, a transitive set $M$ with $\kappa\in M$ and an elementary embedding $j:M\to H_\theta$ such that $j(\kappa)=\kappa$ and $\text{crit }j<\kappa$.
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=== Partition Property ===
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A cardinal $\kappa$ is '''Jónsson''' iff the [[partition property]] $\kappa\rightarrow [\kappa]_\kappa^{<\omega}$ holds, i.e. that given any function $f:[\kappa]^{<\omega}\to\kappa$ we can find a subset $H\subseteq\kappa$ such that $f"[H]^n\neq\kappa$ for every $n<\omega$. <cite>Kanamori2003:HigherInfinite</cite>
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=== Weak Substructure Characterization ===
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A cardinal $\kappa$ is '''Jónsson''' iff Given any $A$ there exists an elementary substructure $\langle X,\in, X\cap A\rangle\prec\langle V_\kappa,\in,A\rangle$ with $|X|=\kappa$ and $X\cap\kappa\neq\kappa$.
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=== Strong Substructure Characterization ===
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A cardinal $\kappa$ is '''Jónsson''' iff any first-order structure with universe of cardinality $\kappa$ has a proper elementary substructure with universe also having cardinality $\kappa$. <cite>Kanamori2003:HigherInfinite</cite>
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=== Embedding Characterization ===
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A cardinal $\kappa$ is '''Jónsson''' iff for every $\theta>\kappa$ there exists a transitive set $M$ with $\kappa\in M$ and an elementary embedding $j:M\to H_\theta$ such that $j(\kappa)=\kappa$ and $\text{crit }j<\kappa$.
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Interestingly, there only needs to be one such $\theta$, and then Jónssonness holds for $\kappa$. Therefore, if one such $\theta>\kappa$ has this property, then every $\theta>\kappa$ has this property as well.
  
 
== Properties ==
 
== Properties ==
  
All the following facts can be found in Kanamori (2009).
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All the following facts can be found in <cite>Kanamori2003:HigherInfinite</cite>:
  
* $\omega$ is not Jónsson.
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* $\aleph_0$ is not Jónsson.
 
* If $\kappa$ isn't Jónsson then neither is $\kappa^+$.
 
* If $\kappa$ isn't Jónsson then neither is $\kappa^+$.
 
* If $2^\kappa=\kappa^+$ then $\kappa^+$ isn't Jónsson.
 
* If $2^\kappa=\kappa^+$ then $\kappa^+$ isn't Jónsson.
* If $\kappa$ is regular then $\kappa^+$ isn't Jónsson.
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* If $\kappa$ is regular then $\kappa^+$ isn't Jónsson (therefore $\kappa^{++}$ is never Jónsson, and if $\kappa$ is weakly inaccessible then $\kappa^+$ is never Jónsson).
* A singular limit of measurables is Jónsson.
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* A singular limit of [[measurable|measurables]] is Jónsson.
* The least Jónsson is either weakly inaccessible or has cofinality $\omega$.
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* The least Jónsson is either [[inaccessible|weakly inaccessible]] or has cofinality $\omega$.
 
* $\aleph_{\omega+1}$ is not Jónsson.
 
* $\aleph_{\omega+1}$ is not Jónsson.
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It is still an open question as to whether or not there is some known large cardinal axiom that implies the consistency of $\aleph_\omega$ being Jónsson.
  
 
== Relations to other large cardinal notions ==
 
== Relations to other large cardinal notions ==
  
 
Jónsson cardinals have a lot of consistency strength:
 
Jónsson cardinals have a lot of consistency strength:
* Jónsson cardinals are equiconsistent with Ramsey cardinals (Mitchell, 1999).
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* Jónsson cardinals are equiconsistent with [[Ramsey]] cardinals. <cite>Mitchell1997:JonssonErdosCoreModel</cite>
* The existence of a Jónsson cardinal $\kappa$ implies the existence of $x^\sharp$ for every $x\in V_\kappa$.
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* The existence of a Jónsson cardinal $\kappa$ implies the existence of [[Zero sharp|$x^\sharp$]] for every $x\in V_\kappa$ (and therefore for every real number $x$, because $\kappa$ is uncountable).
  
But in terms of ''actual'' strength, they're (ostensibly) quite weak:
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But in terms of size, they're (ostensibly) quite small:
* A Jónsson cardinal need not be regular.
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* A Jónsson cardinal need not be regular (assuming the consistency of a [[measurable]] cardinal).
* Every Ramsey cardinal is inaccessible Jónsson.
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* Every Ramsey cardinal is inaccessible and Jónsson. <cite>Kanamori2009:HigherInfinite</cite>
* Every (weakly) inaccessible Jónsson is (weakly) hyper-Mahlo (Shelah, 1994).
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* Every weakly inaccessible Jónsson is [[Mahlo|weakly hyper-Mahlo]]. <cite>Shelah1994:CardinalArithmetic</cite>
  
It's an open question whether or not every inaccessible Jónsson is weakly compact (Welch, 1998).
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It's an open question whether or not every inaccessible Jónsson cardinal is [[weakly compact]].
  
 
== Jónsson successors of singulars ==
 
== Jónsson successors of singulars ==
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== Jónsson cardinals and the core model ==
 
== Jónsson cardinals and the core model ==
  
Assuming there is no inner model with a Woodin cardinal then:
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In 1998, Welch proved many interesting facts about Jónsson cardinals and the core model these can be found in <cite>Welch1998:InnerModels</cite>.
* Weak covering holds at every Jónsson cardinal, i.e. that $\kappa^{+K}=\kappa^+$ for every Jónsson cardinal (Welch, 1998).
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Assuming there is no inner model with a [[Woodin]] cardinal then:
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* Weak covering holds at every Jónsson cardinal, i.e. that $\kappa^{+K}=\kappa^+$ for every Jónsson cardinal.
 
* If $\kappa$ is regular Jónsson then the set of regular $\alpha<\kappa$ satisfying weak covering is stationary in $\kappa$ (Welch, 1998).
 
* If $\kappa$ is regular Jónsson then the set of regular $\alpha<\kappa$ satisfying weak covering is stationary in $\kappa$ (Welch, 1998).
  
If we assume that there's no sharp for a strong cardinal (known as $0^\P$ doesn't exist) then:
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If we assume that there's no sharp for a [[strong]] cardinal (known as $0^\P$ doesn't exist) then:
* For a Jónsson cardinal $\kappa$, $A^\sharp$ exists for every $A\subseteq\kappa$ (Welch, 1998).
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* For a Jónsson cardinal $\kappa$, [[Zero sharp|$A^\sharp$]] exists for every $A\subseteq\kappa$ (Welch, 1998).
  
 
== References ==
 
== References ==
  
* Kanamori, Akihiro: "The Higher Infinite", second edition, 2009
 
 
* Koepke, Hans-Dieter & Koepke, Peter: "On the Consistency Strength of 'Accessible' Jónsson Cardinals and of the Weak Chang Conjecture", Annals of Pure and Applied Logic 25: 233-261, 1983
 
* Koepke, Hans-Dieter & Koepke, Peter: "On the Consistency Strength of 'Accessible' Jónsson Cardinals and of the Weak Chang Conjecture", Annals of Pure and Applied Logic 25: 233-261, 1983
 
* Mitchell, William: "Jonsson Cardinals, Erdos Cardinals and the Core Model", Journal of Symbolic Logic 64(3): 1065-1086, 1999
 
* Mitchell, William: "Jonsson Cardinals, Erdos Cardinals and the Core Model", Journal of Symbolic Logic 64(3): 1065-1086, 1999
* Shelah, Saharon: "Cardinal Arithmetic", Oxford Logic Guides, vol. 29, Oxford Univ. Press, London and New York, 1994
 
 
* Tryba, Jan: "On Jónsson cardinals with uncountable cofinality", Israel Journal of Mathematics 49(4): 315-324, 1984
 
* Tryba, Jan: "On Jónsson cardinals with uncountable cofinality", Israel Journal of Mathematics 49(4): 315-324, 1984
* Welch, Philip: "Some Remarks on the Maximality of Inner Models", Logic Colloquium 1998, available at [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.41.7037&rep=rep1&type=pdf]
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{{References}}

Revision as of 07:24, 14 November 2017

Jónsson cardinals are named after Bjarni Jónsson, a student of Tarski working in universal algebra. In 1962, he asked whether or or not every algebra of cardinality $\kappa$ has a proper subalgebra of the same cardinality. The cardinals $\kappa$ that satisfy this property are now called Jónsson cardinals.

An algebra of cardinality $\kappa$ is simply a set $A$ of cardinality $\kappa$ and finitely many functions (each with finitely many inputs) $f_0,f_1...f_n$ for which $A$ is closed under every such function. A subalgebra of such an algebra is a set $B\subseteq A$ such that $B$ is closed under

Equivalent Definitions

There are several equivalent definitions to Jónsson cardinals cardinals.

Partition Property

A cardinal $\kappa$ is Jónsson iff the partition property $\kappa\rightarrow [\kappa]_\kappa^{<\omega}$ holds, i.e. that given any function $f:[\kappa]^{<\omega}\to\kappa$ we can find a subset $H\subseteq\kappa$ such that $f"[H]^n\neq\kappa$ for every $n<\omega$. [1]

Weak Substructure Characterization

A cardinal $\kappa$ is Jónsson iff Given any $A$ there exists an elementary substructure $\langle X,\in, X\cap A\rangle\prec\langle V_\kappa,\in,A\rangle$ with $|X|=\kappa$ and $X\cap\kappa\neq\kappa$.

Strong Substructure Characterization

A cardinal $\kappa$ is Jónsson iff any first-order structure with universe of cardinality $\kappa$ has a proper elementary substructure with universe also having cardinality $\kappa$. [1]

Embedding Characterization

A cardinal $\kappa$ is Jónsson iff for every $\theta>\kappa$ there exists a transitive set $M$ with $\kappa\in M$ and an elementary embedding $j:M\to H_\theta$ such that $j(\kappa)=\kappa$ and $\text{crit }j<\kappa$.

Interestingly, there only needs to be one such $\theta$, and then Jónssonness holds for $\kappa$. Therefore, if one such $\theta>\kappa$ has this property, then every $\theta>\kappa$ has this property as well.

Properties

All the following facts can be found in [1]:

  • $\aleph_0$ is not Jónsson.
  • If $\kappa$ isn't Jónsson then neither is $\kappa^+$.
  • If $2^\kappa=\kappa^+$ then $\kappa^+$ isn't Jónsson.
  • If $\kappa$ is regular then $\kappa^+$ isn't Jónsson (therefore $\kappa^{++}$ is never Jónsson, and if $\kappa$ is weakly inaccessible then $\kappa^+$ is never Jónsson).
  • A singular limit of measurables is Jónsson.
  • The least Jónsson is either weakly inaccessible or has cofinality $\omega$.
  • $\aleph_{\omega+1}$ is not Jónsson.

It is still an open question as to whether or not there is some known large cardinal axiom that implies the consistency of $\aleph_\omega$ being Jónsson.

Relations to other large cardinal notions

Jónsson cardinals have a lot of consistency strength:

  • Jónsson cardinals are equiconsistent with Ramsey cardinals. [2]
  • The existence of a Jónsson cardinal $\kappa$ implies the existence of $x^\sharp$ for every $x\in V_\kappa$ (and therefore for every real number $x$, because $\kappa$ is uncountable).

But in terms of size, they're (ostensibly) quite small:

  • A Jónsson cardinal need not be regular (assuming the consistency of a measurable cardinal).
  • Every Ramsey cardinal is inaccessible and Jónsson. [3]
  • Every weakly inaccessible Jónsson is weakly hyper-Mahlo. [4]

It's an open question whether or not every inaccessible Jónsson cardinal is weakly compact.

Jónsson successors of singulars

As mentioned above, $\aleph_{\omega+1}$ is not Jónsson (this is due to Shelah). The question is then if it's possible for any successor of a singular cardinal to be Jónsson. Here is a (non-exhaustive) list of things known:

  • If $0\neq\gamma<|\eta|$ then $\aleph_{\eta+\gamma+1}$ is not Jónsson (Tryba, 1984).
  • If there exists a Jónsson successor of a singular then $0^\dagger$ exists (Donder & Koepke, 1983).

Jónsson cardinals and the core model

In 1998, Welch proved many interesting facts about Jónsson cardinals and the core model these can be found in [5]. Assuming there is no inner model with a Woodin cardinal then:

  • Weak covering holds at every Jónsson cardinal, i.e. that $\kappa^{+K}=\kappa^+$ for every Jónsson cardinal.
  • If $\kappa$ is regular Jónsson then the set of regular $\alpha<\kappa$ satisfying weak covering is stationary in $\kappa$ (Welch, 1998).

If we assume that there's no sharp for a strong cardinal (known as $0^\P$ doesn't exist) then:

  • For a Jónsson cardinal $\kappa$, $A^\sharp$ exists for every $A\subseteq\kappa$ (Welch, 1998).

References

  • Koepke, Hans-Dieter & Koepke, Peter: "On the Consistency Strength of 'Accessible' Jónsson Cardinals and of the Weak Chang Conjecture", Annals of Pure and Applied Logic 25: 233-261, 1983
  • Mitchell, William: "Jonsson Cardinals, Erdos Cardinals and the Core Model", Journal of Symbolic Logic 64(3): 1065-1086, 1999
  • Tryba, Jan: "On Jónsson cardinals with uncountable cofinality", Israel Journal of Mathematics 49(4): 315-324, 1984

References

  1. Mitchell, William J. Jónsson Cardinals, Erdős Cardinals, and the Core Model. J Symbol Logic , 1997. www   bibtex
  2. 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
  3. Shelah, Saharon. Cardinal Arithmetic. Oxford Logic Guides 29, 1994. bibtex
  4. Welch, Philip. Some remarks on the maximality of Inner Models. Logic Colloquium , 1998. www   bibtex
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