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 $\kappa$ 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 that algebra is a set $B\subseteq A$ which $B$ is closed under each $f_k$ for $k\leq n$.
Contents
Equivalent Definitions
There are several equivalent definitions of Jónsson 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 $H\subseteq\kappa$ of order type $\kappa$ such that $f``[H]^n\neq\kappa$ for every $n<\omega$. [1]
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 $X\cap\kappa\neq\kappa$.
- A cardinal $\kappa$ is Jónsson iff any 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 $\text{crit }j<\kappa$, iff for every $\theta>\kappa$ there exists a transitive set $M$ with $\kappa\in M$ and an elementary embedding $j:M\to V_\theta$ such that $j(\kappa)=\kappa$ and $\text{crit} j<\kappa$.
Interestingly, if one such $\theta>\kappa$ has this property, then every $\theta>\kappa$ has this property as well.
Abstract Algebra Characterization
In terms of abstract algebra, $\kappa$ is Jónsson iff any algebra $A$ of size $\kappa$ has a proper subalgebra of $A$ of size $\kappa$.
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. [5]
- If there exists a Jónsson successor of a singular cardinal then $0^\dagger$ exists. [6]
Jónsson cardinals and the core model
In 1998, Welch proved many interesting facts about Jónsson cardinals and the core model that can be found in [7]. 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 $\kappa$.
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 $A\subseteq\kappa$.
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
- Tryba, Jan. On Jónsson cardinals with uncountable cofinality. Israel Journal of Mathematics 49(4), 1983. bibtex
- Donder, Hans-Dieter and Koepke, Peter. On the Consistency Strength of 'Accessible' Jónsson Cardinals and of the Weak Chang Conjecture. Annals of Pure and Applied Logic , 1998. www DOI bibtex
- Welch, Philip. Some remarks on the maximality of Inner Models. Logic Colloquium , 1998. www bibtex