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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.
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 $k\leq n$.
There are several equivalent definitions of Jónsson cardinals.
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)
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)
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.
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.
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.
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:
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: