It was mentioned after Theorem 30.27 in Kanamori's Higher Infinite that Woodin constructed a model of $DC$ + there exists unboundedly many many $\kappa<\Theta$ such that $\kappa \to (\kappa)^\kappa_{\alpha} \ \forall \alpha<\kappa$ and there exists a non-principal ultrafilter on $\omega$. In particular, this model is not a model of $AD$.

What is the reference of this? Or even better for those who know the proof, what does the proof look like? Thanks!

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    $\begingroup$ You start with determinacy in $L(\mathbb R)$ and add an ultrafilter by a forcing that adds no sets of ordinals, such as $\mathcal P(\omega)/\mathrm{Fin}$. $\endgroup$ – Andrés E. Caicedo Aug 30 '18 at 16:00
  • $\begingroup$ The reference you want is "A barren extension" by Henle, Mathias and Woodin. I'll try to post a brief answer later today. $\endgroup$ – Andrés E. Caicedo Aug 30 '18 at 16:35
  • $\begingroup$ To add to the comment of @Andrés, since $\mathcal P(\omega)/\rm fin$ is also $\sigma$-closed, it preserves $\sf DC$. $\endgroup$ – Asaf Karagila Aug 30 '18 at 16:35

The key reference for this is

MR0799042 (87d:03141). Henle, J. M.; Mathias, A. R. D.; Woodin, W. Hugh. A barren extension. In Methods in mathematical logic (Caracas, 1983), C. A. Di Prisco, editor, 195–207, Lecture Notes in Math., 1130, Springer, Berlin, 1985.

There, Henle, Mathias, and Woodin start with $L(\mathbb R)$ under the assumption of determinacy (and $\mathsf{DC}$), and force with $\mathcal P(\omega)/\mathrm{Fin}$; they refer to the resulting model as "the Hausdorff extension".

They use $\mathsf{ZF}+\omega\to(\omega)^\omega$ to prove that the Hausdorff extension is barren, meaning that every map from an ordinal into the ground model was already in the ground model. They also show (under $\mathsf{AD}+V=L(\mathbb R)$) that all strong partition cardinals below $\Theta$ are preserved in the extension. Easily, the extension also preserves $\mathsf{DC}$.

On the other hand, the forcing adds a Ramsey ultrafilter on $\omega$ (in particular, unboundedly many strong partition cardinals below $\Theta$ is not enough to ensure all sets all reals are Lebesgue measurable).

  • $\begingroup$ (Let me know if you need a copy of the Henle-Mathias-Woodin paper.) $\endgroup$ – Andrés E. Caicedo Aug 30 '18 at 17:25
  • $\begingroup$ Great! Thanks! I managed to find its internet existence. This is helpful! $\endgroup$ – Jing Zhang Aug 30 '18 at 18:16

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