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

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

William Mitchell proved, with the help of the Dodd-Jensen core model that the consistency of the existence of a Jónsson cardinal implies the consistency of the existence of a Ramsey cardinal, so that the existence of Jónsson cardinals and the existence of Ramsey cardinals are equiconsistent.

Using the axiom of choice, a lot of small cardinals (the

Results like this need the axiom of choice, however: The axiom of determinacy does imply that for every positive natural number n, the cardinal

Here an algebra means a 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.

The existence of Jónsson functions shows that if algebras are allowed to have infinitary operations, then there are no analogues of Jónsson cardinals.

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