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I0 and rank-into-rank axioms

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Abstract

This is a survey about I0 and rank-into-rank axioms, with some previously unpublished proofs.

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Notes

  1. Admittedly, this is a very rough classification, as for example indescribable cardinals are difficult to insert in this narrative.

  2. In fact, \(S^{\lambda ^+}_\omega \) is \(\omega \)-stationary, that is, it intersects all the \(\omega \)-clubs, i.e., the sets that are cofinal in \(\lambda ^+\) and closed under supremum of \(\omega \)-sequences.

  3. This is not a standard definition. For different authors, “\(\omega \)-huge” can mean different things.

  4. This Proposition and Proposition 5.5 are in collaboration with Alessandro Andretta.

  5. With \(\mathbb {R}\)-perfect set we intend that \(2^\omega \) can be continuously embedded in Z, a weaker property than to be a perfect set. Therefore this Lemma is now incorporated in 6.8.

  6. The proof in [39] is more convoluted, as it proves that \(V_{\lambda +1}\), if in \(L(V[G]_{\lambda +1})\) and \(\mathbb {U}(j)\)-representable, is a perfect set, and this brings a contradiction. By 6.8 we immediately have that \(V_{\lambda +1}\) is perfect.

  7. It is the product with support the Easton ideal, i.e., the sets that are bounded below every inaccessible cardinal.

  8. This rather clumsy statement comes from the fact that the proof uses the fact that some \(Z_S\) defined from the stationary set S is \(\mathbb {U}(j)\)-representable. If all the sets in some \(L_\delta (V_{\lambda +1})\), with \(\delta \) limit, are \(\mathbb {U}(j)\)-representable, this implies that we cannot find two disjoint stationary sets in \(L_\delta (V_{\lambda +1})\). If all the sets are \(\mathbb {U}(j)\)-representable, this means that the club filter is an ultrafilter on cofinality \(\omega \).

  9. Recall that \(\kappa \rightarrow (\kappa )^\alpha _\beta \) is that if \(\pi :\{\sigma \subseteq \kappa :{{\mathrm{ot}}}(\sigma )=\alpha \}\rightarrow \beta \) then there is a set \(H\subseteq \kappa \) of cardinality \(\kappa \) homogeneous for \(\pi \), i.e., \(|\{\pi (\sigma ):\sigma \subseteq H,\ {{\mathrm{ot}}}(\sigma )=\alpha \}|=1\).

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Acknowledgements

The paper was written under the Italian program “Rita Levi Montalcini 2013”, but it is a collection of notes, thoughts, sketches and ideas that started in the Summer Semester of 2008 at Berkeley, propelled by a course on “Beyond I0” by Woodin. With nine years in the making it is therefore a hard task to thank all the people that helped me writing this paper, as almost all the people I interacted with in my academic career has been an inspiration for this, directly or indirectly. Knowing that this is just a small percentage of the whole picture, I would like to thank first of all Hugh Woodin, the demiurge of the I0 world, that welcomed me in Berkeley when I was still a student and continued to be a creative influence during the years; Alessandro Andretta, for his steady support in all this years, and because of his (decisive) insistence for this survey to be written. I would also like to thank Sy Friedman and Liuzhen Wu, that helped to push my research in an unexpected direction, and Scott Cramer and Xianghui Shi, for the many, many discussions on I0 and similars. Special thanks also for Luca Motto Ros, that helped me in shaping this paper.

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  1. Università degli Studi di Udine, via delle Scienze, 206 33100, Udine, UD, Italy

    Vincenzo Dimonte

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Dimonte, V. I0 and rank-into-rank axioms. Boll Unione Mat Ital 11, 315–361 (2018). https://doi.org/10.1007/s40574-017-0136-y

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