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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 $k\leq n$.

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

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 $\kappa$.


All the following facts can be found in (Kanamori, 2009):

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:

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

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:

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:

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


  1. Kanamori, A. (2009). The higher infinite (Second, p. xxii+536). Springer-Verlag.
  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.
  6. Welch, P. (1998). Some remarks on the maximality of Inner Models. Logic Colloquium.
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