Jónsson cardinal

• Equivalent Definitions
  • Partition Property
  • Substructure Characterization
  • Embedding Characterization
  • Abstract Algebra Characterization
• Properties
  • Relations to other large cardinal notions
  • Jónsson successors of singulars
• Jónsson cardinals and the core model

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 $k\leq n$.

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 $n<\omega$. (Kanamori, 2009)

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 $\kappa$. (Kanamori, 2009)

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 (Kanamori, 2009):

• $\aleph_0$ 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. (Mitchell, 1997)
• 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. (Kanamori, 2009)
• Every weakly inaccessible Jónsson is weakly hyper-Mahlo. (Shelah, 1994)

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, 1983)
• If there exists a Jónsson successor of a singular cardinal then $0^\dagger$ exists. (Donder & Koepke, 1998)

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 (Welch, 1998). 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

1. Kanamori, A. (2009). The higher infinite (Second, p. xxii+536). Springer-Verlag. https://link.springer.com/book/10.1007%2F978-3-540-88867-3
2. Mitchell, W. J. (1997). Jónsson Cardinals, Erdős Cardinals, and the Core Model. J. Symbol Logic.
3. Shelah, S. (1994). Cardinal Arithmetic. Oxford Logic Guides, 29.
4. Tryba, J. (1983). On Jónsson cardinals with uncountable cofinality. Israel Journal of Mathematics, 49(4).
5. Donder, H.-D., & Koepke, P. (1998). On the Consistency Strength of Áccessible\’Jónsson Cardinals and of the Weak Chang Conjecture. Annals of Pure and Applied Logic. https://doi.org/10.1016/0168-0072(83)90020-9
6. Welch, P. (1998). Some remarks on the maximality of Inner Models. Logic Colloquium. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.41.7037&rep=rep1&type=pdf

Main library