Jónsson cardinal

In set theory, a Jónsson cardinal (named after Bjarni Jónsson) is a certain kind of large cardinal number.

An uncountable cardinal number κ is said to be "Jónsson" if for every function "f": [κ] ^<ω → κ there is a set "H" of order type κ such that for each "n", "f" restricted to "n"-element subsets of "H" omits at least one value in κ.

Every Rowbottom cardinal is Jónsson. By a theorem of Kleinberg, the theories ZFC + “there is a Rowbottom cardinal” and ZFC + “there is a Jónsson cardinal” are equiconsistent.

In general, Jónsson cardinals need not be large cardinals in the usual sense: they can be singular. Using the axiom of choice, a lot of small cardinals (the $aleph\_n$, for instance) can be proved to be not Jónsson. Results like this need the axiom of choice, however: The axiom of determinacy does imply that for every positive natural number "n", the cardinal $aleph\_n$ is Jónsson.

A Jónsson algebra is an algebra with no proper subalgebras of the same cardinality. (Here an algebra meansa model for a language with a countable number of function symbols, in other words a set with a countable number of functions from finite products of the set to itself.) A cardinal is a Jónsson cardinal if and only if there are no Jónsson algebras of that cardinality.

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