Luminy – Hugh Woodin: Ultimate L (I)
The XI International Workshop on Set Theory took place October 4-8， 2010. It was hosted by the CIRM， in Luminy， France. I am very glad I was invited， since it was a great experience: The Workshop has a tradition of excellence， and this time was no exception， with several very nice talks. I had the chance to give a talk (available here) and to interact with the other participants. There were two mini-courses， one by Ben Miller and one by Hugh Woodin. Ben has made the slides of his series available at his website.
What follows are my notes on Hugh’s talks. Needless to say， any mistakes are mine. Hugh’s talks took place on October 6， 7， and 8. Though the title of his mini-course was “Long extenders， iteration hypotheses， and ultimate L”， I think that “Ultimate L” reflects most closely the content. The talks were based on a tiny portion of a manuscript Hugh has been writing during the last few years， originally titled “Suitable extender sequences” and more recently，“Suitable extender models” which， unfortunately， is not currently publicly available.
The general theme is that appropriate extender models for superpactness should provably be an ultimate version of the constructible universe L. The results discussed during the talks aim at supporting this idea.
Ultimate L
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I
Let δ be superpact. The basic problem that concerns us is whether there is an L-like inner model N\subseteq V with δ superpact in N.
Of course， the shape of the answer depends on what we mean by “L-like”. There are several possible ways of making this nontrivial. Here， we only adopt the very general requirement that the superpactness of δ in N should “directly trace back” to its superpactness in V.
Recall:
We use P_δ(X) to denote the set \{a\subseteq X\mid |a|amp;amp;amp;amp;lt;δ\}.
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