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%20property.md)
$\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 $n<\omega$.
{% cite Kanamori2009 %}
### 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 $\kappa$.
{% cite Kanamori2009 %}
### 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
{% cite Kanamori2009 %}:
- $\aleph_0$ 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](Measurable.md "Measurable")
is Jónsson.
- The least Jónsson is either [](Inaccessible.md)
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](Ramsey.md "Ramsey")
cardinals. {% cite Mitchell1997 %}
- The existence of a Jónsson cardinal $\kappa$ implies the existence
of
<a href="Zero_sharp" class="mw-redirect" title="Zero sharp">$x^\sharp
lt;/a>
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](Measurable.md "Measurable")
cardinal).
- Every Ramsey cardinal is inaccessible and Jónsson.
{% cite Kanamori2009 %}
- Every weakly inaccessible Jónsson is [](Mahlo.md).
{% cite Shelah1994 %}
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. {% cite Tryba1983 %}
- If there exists a Jónsson successor of a singular cardinal then
<a href="Zero_dagger" class="mw-redirect" title="Zero dagger">$0^\daggerlt;/a>
exists. {% cite Donder1998 %}
## 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
{% cite Welch1998 %}. Assuming there is no inner model
with a
[Woodin](Woodin.md "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](Strong.md "Strong")
cardinal (known as $0^\P$ doesn't exist) then:
- For a Jónsson cardinal $\kappa$,
<a href="Zero_sharp" class="mw-redirect" title="Zero sharp">$A^\sharplt;/a>
exists for every $A\subseteq\kappa$.