# 3rd Münster conference on

# inner model theory, the core model induction, and hod mice

## Institut für Mathematische Logik, WWU Münster,

## July 20 — 31, 2015

Once more, this conference will draw together researchers and advanced students with an interest in inner model theory, in order to communicate and further explore this recent work. There will be courses and single talks.

We will meet Monday–Friday, with 2 hours of lecture in the morning and 2 hours of lecture in the early afternoon. This will leave ample time for problem sessions, informal seminars, and other interactions in the late afternoons and evenings. The week July 20 — 24, 2014, will be the week when the formal part of the conference will take place; the organizers will decide beforehand about speakers and talks for the 1st week. There will be a second week, July 27 — 31, 2014, which will be much less formal; speakers and talks for the 2nd week will be decided about during the first week.

The conference will take place at the Department of Mathematics and Computer Science, Univ. of Muenster, Orléans-Ring 10, 48149 Muenster, FRG, lecture halls N2 and N3. We also have lecture hall N2 available for the week June 13 — 17 for pre–conference activities.

The conference organizers gratefully acknowledge financial support from the Marianne and Dr. Horst Kiesow-Stiftung, Frankfurt a.M., from the DVMLG (Deutsche Vereinigung für mathematische Logik und Grundlagenforschung), as well as from the NSF (National Science Foundation).

## Schedule:

Mo, July 20 | Tue, July 21 | We, July 22 | Thu, July 23 | Fr, July 24 | |

9:30–10:45 | Yizheng Zhu | Trevor Wilson | Nam Trang | Ralf Schindler | Martin Zeman |

11:15–12:30 | Gunter Fuchs | Farmer Schlutzenberg | Daniel Rodrigues | Steve Jackson | Chris Le Sueur |

14:30–15:45 | John Steel 1 | Hugh Woodin 1 | Hugh Woodin 3 | John Steel 3 | Philipp Schlicht |

16:45–17:30 | John Steel 2 | Hugh Woodin 2 | Hugh Woodin 4 | John Steel 4 | N.N. |

17:30–∞ | Problems and Discussions | Problems and Discussions | Problems and Diskussions | Problems and Discussions | free |

Mo, July 27 | Tue, July 28 | We, July 29 | Thu, July 30 | Fr, July 31 | |

9:30–11:00 | N.N. | N.N. | N.N. | N.N. | N.N. |

11:30–12:30 | N.N. | N.N. | N.N. | N.N. | N.N. |

14:30–16:00 | N.N. | N.N. | N.N. | N.N. | free |

16:30–17:30 | N.N. | N.N. | N.N. | N.N. | free |

17:30–∞ | Problems and Discussions | Problems and Discussions | Problems and Diskussions | Problems and Discussions | free |

## Talks and abstracts:

- Gunter Fuchs: Geology, inner models and the solid core. I will present some results due to Ralf Schindler and myself, isolating situations in which certain models of set theory that arise in the context of set theoretic geology turn out to be fine structural. For example, if the universe is constructible from a set and there is an inner model with a Woodin cardinal, then the mantle, i.e., the intersection of all inner models of which the universe is a forcing extension by set forcing, is fine structural. A similar result concerns the concept of the solid core, which is the collection of all sets that are constructible from a set of ordinals that is solid, meaning that it cannot be added to an inner model by set forcing. Again, the solid core is fine structural if there is an inner model with a Woodin cardinal.
- Steve Jackson: Some closure properties of pointclasses under AD.
- Daniel Rodriguez: L(R,μ) is unique. We show that under various appropriate hypotheses there is only one determinacy model of the form L(R,μ) in which mu is a supercompact measure on P
_{ω1}(R). - Grigor Sargsyan: The Largest Suslin Axiom. We will state the Largest Suslin Axiom. Show its consistency relative to large cardinals. Show how to derive it from combinatorial principles (in the presence of mild large cardinals), and if time permits, make some comments about how to go beyond. (Cancelled.)
- Ralf Schindler: AD from ℵ
_{ω}is a strong limit and 2^{ℵω}> ℵ_{ω1}. Using a cheat, we get an inner model with ω_{1}Woodin cardinals from said hypothesis and also from the assumption that κ is a limit of cofinality ω_{1}s.t. the set of μ < κ with 2^{μ}= μ^{+}is stationary and costationary. This is joint with Gabriel Fernandes. - Philipp Schlicht: M
_{1}and infinite time machines. A subset of an ordinal is recognizable if it is the unique subset for which an infinite time machine has a certain output. We consider reals as inputs to infinite time machines with finitely many ordinal parameters, and define the recognizable hull by iterating relative recognizability for reals. In joint work with Merlin Carl, we connect this with M_{1}by showing that a real is in the recognizable hull if and only if it is in M_{1}. - Farmer Schlutzenberg: Fine structure from normal iterability. For short extender premice, the solidity of the standard parameter and related fine structural properties, follow from normal iterability. That is, let k< ω and let M be a k-sound, (k, ω
_{1}+1)-iterable premouse. Then M is k+1-solid and k+1-universal, degree k+1-condensation holds for M, and if k>0 then M is Dodd-sound. Further, any (0, ω_{1}+1)$-iterable pseudo-premouse is a premouse. I will explain at least the main ideas in the proofs of these results. (The results hold for Mitchell-Steel indexing, permitting extenders of superstrong type to appear on the extender sequence.) - John Steel: Normalizing iteration trees and comparing iteration strategies.
- Chris Le Sueur: Determinacy a bit beyond co-analytic. It is quite well-known result of Martin that the existence of a measurable cardinal is enough to prove the determinacy of all boldface Π
^{1}_{1}sets. The argument nicely modifies to get the determinacy of all lightface Π^{1}_{1}sets from the existence of O^{♯}. This argument has since been pushed to go beyond Π^{1}_{1}, and I will discuss how this is done. In doing so, I’ll introduce a generalised effective descriptive set theory suitable for studying uncountable spaces, and describe a ramified forcing (akin to Cohen’s original conception) that allows class forcing over weak models. - Nam Trang: Square in hod mice and applications. We discuss ideas for proving square in hod mice up to lsa hod mice. We describe how the straightforward adaptation of the Schimmerling-Zeman construction fails for hod mice and how to overcome it. Applications of this type of theorems include improving lower-bound consistency strength of various principles. For instance, I prove that PFA implies the sharp for the minimal model of “AD
_{R}+ Θ is strong” exists. We hope to build on this to construct a model of “AD^{+}+ LSA” from principles like PFA. - Trevor Wilson: Determinacy from strong compactness of ω
_{1}. Determinacy axioms are well-known to imply large cardinal properties of ω_{1}such as measurability and certain degrees of strong compactness and supercompactness. Conversely, from large cardinal properties of ω_{1}one can obtain inner models of determinacy axioms using the core model induction. This approach leads to some equiconsistencies between determinacy axioms and their consequences in terms of degrees of strong compactness of ω_{1}. For a set X, we say that ω_{1}is X-strongly compact if there is a fine, countably complete measure on P_{ω1}(X). We obtain the following equiconsistencies modulo ZF plus DC: (1) the P(ω_{1})-strong compactness of ω_{1}is equiconsistent with AD, and (2) the P(R)-strong compactness of ω_{1}is equiconsistent with AD_{R}(plus DC). This talk is based on joint work with Nam Trang. - Hugh Woodin: The Ultimate-L Conjecture. I discuss progress on proving the Ultimate-L Conjecture focusing on two obstructions. The first obstruction concerns comparison by least disagreement and the second concerns the necessary failure of amenable or even weakly amenable levels. The latter is just the necessary failure of weak square principles at singular cardinals translated into fine-structural terms. Both these obstructions provide key clues as to how to pass the “ω-barrier”.
- Yizheng Zhu: The higher sharp. We establish the descriptive set theoric representation of the mouse M
_{n}^{#}, which is called 0^{(n+1)#}. At even levels, 0^{2n#}is the higher level analog of Kleene’s O; at odd levels, 0^{2n+1#}is the unique iterable remarkable level-2n+1 blueprint. notes taken by rds - Martin Zeman: The true core model with a Woodin cardinal. We show that in certain universes one can construct the true core model using Steel’s CMIP construction. The model will be the true core model in the sense that it will satisfy the rigidity theorem, generic absoluteness, and weak covering. This is a joint work with Grigor Sargsyan.