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 $\kappa$ | + | [[Category:Large cardinal axioms]] |
+ | [[Category:Partition property]] | ||
+ | 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 $n<\omega$. <cite>Kanamori2003:HigherInfinite</cite> | ||
+ | |||
+ | === 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 $\kappa$. <cite>Kanamori2003:HigherInfinite</cite> | ||
+ | |||
+ | === 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 == | == Properties == | ||
− | All the following facts can be found in | + | All the following facts can be found in <cite>Kanamori2003:HigherInfinite</cite>: |
− | * $\ | + | * $\aleph_0$ 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. | + | * 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. | + | * A singular limit of [[measurable|measurables]] is Jónsson. |
− | * The least Jónsson is either weakly inaccessible or has cofinality $\omega$. | + | * The least Jónsson is either [[inaccessible|weakly inaccessible]] or has cofinality $\omega$. |
* $\aleph_{\omega+1}$ is not Jónsson. | * $\aleph_{\omega+1}$ 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 == | == 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 | + | * 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 $x\in V_\kappa$. | + | * 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 | + | But in terms of size, they're (ostensibly) quite small: |
− | * A Jónsson cardinal need not be regular. | + | * A Jónsson cardinal need not be regular (assuming the consistency of a [[measurable]] cardinal). |
− | * Every Ramsey cardinal is inaccessible Jónsson. | + | * Every Ramsey cardinal is inaccessible and Jónsson. <cite>Kanamori2009:HigherInfinite</cite> |
− | * Every | + | * 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 | + | It's an open question whether or not every inaccessible Jónsson cardinal is [[weakly compact]]. |
== Jónsson successors of singulars == | == Jónsson successors of singulars == | ||
Line 46: | Line 59: | ||
== Jónsson cardinals and the core model == | == Jónsson cardinals and the core model == | ||
− | Assuming there is no inner model with a Woodin cardinal then: | + | 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 | + | 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 $\kappa$ 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: | + | 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). | + | * For a Jónsson cardinal $\kappa$, [[Zero sharp|$A^\sharp$]] exists for every $A\subseteq\kappa$ (Welch, 1998). |
== References == | == 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 | * 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 | ||
− | |||
* 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 | ||
− | + | ||
+ | {{References}} |
Revision as of 08:24, 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
Contents
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
- Mitchell, William J. Jónsson Cardinals, Erdős Cardinals, and the Core Model. J Symbol Logic , 1997. www bibtex
- 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
- Shelah, Saharon. Cardinal Arithmetic. Oxford Logic Guides 29, 1994. bibtex
- Welch, Philip. Some remarks on the maximality of Inner Models. Logic Colloquium , 1998. www bibtex