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Is the consistency of the following well-known:

$(*)$: There exists a singular cardinal $\kappa$ such that :

(1) $\kappa$ is a Jonsson cardinal,

(2) $\kappa$ is not a fixed point of the $\aleph-$function, i.e., $\kappa < \aleph_\kappa.$

If we remove the second assumption, then the answer is easy:

(A): Prikry forcing over a measurable cardinal $\kappa,$ turns $\kappa$ into a singular Jonsson cardinal, but $\kappa=\aleph_\kappa$ in such a model.

(B): If $κ$ is a singular limit of measurable cardinals, then $κ$ is Jonsson, but again $\kappa=\aleph_\kappa$ .

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  • $\begingroup$ What happens if $\kappa$ is the limit of $\omega$ measurable cardinals, and you collapse them to be $\aleph_{\omega+n}$, or singularize them and collapse them to be $\aleph_{\omega\cdot n}$? $\endgroup$ – Asaf Karagila Feb 8 '15 at 6:44
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    $\begingroup$ Maybe the paper "On the consistency strength of "accessible" Jonsson cardinals" of Donder and Koepke will be useful for getting a lower bound. They prove there that the existence of such Jonsson cardinal implies the existence of zero dagger. I think that using the current inner models machinery, one can improve their result and get a Woodin cardinal. $\endgroup$ – Yair Hayut Feb 8 '15 at 21:11

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