On successors of Jónsson cardinals

Abstract.

We show that, like singular cardinals, and weakly compact cardinals, Jensen's core model K for measures of order zero [4] calculates correctly the successors of Jónsson cardinals, assuming $O^{Sword}$ does not exist. Namely, if $\kappa$ is a Jónsson cardinal then $\kappa^+ = \kappa^{+K}$, provided that there is no non-trivial elementary embedding $j:K \longrightarrow K$. There are a number of related results in ZFC concerning $\cal{P}(\kappa)$ in V and inner models, for $\kappa$ a Jónsson or singular cardinal.