Conference on

the core model induction and hod mice

Institut für Mathematische Logik, WWU Münster,

July 19 -- August 06, 2010

Organizers: Ralf Schindler (Münster), John Steel (Berkeley)

The core model induction is a method that combines core model theory and descriptive set theory so as to produce logically complicated iteration strategies and scales, via an induction on their logical complexity. It is our most powerful method for obtaining consistency strength lower bounds beyond one Woodin cardinal. In many important cases (e.g. PFA), it seems unlikely one can produce anything like optimal consistency strength lower bounds without a core model induction.

The core model induction method relies heavily on results which connect the hierarchy of scaled pointclasses in determinacy models (more precisely, models of ZF + AD+) with iteration strategies and mice. The basic theory needed here is very much unfinished, but recently there has been some exciting progress. For example, Grigor Sargsyan has shown that if AD+ + V=L(P(R)) holds and there is no model of ADR + "θ is regular" containing all reals and ordinals, then HOD is a hod mouse. Any hod mouse is an ordinary L[E]-mouse up to its own ω1, so Sargsyan's theorem also proves that the well-known Mouse Set Conjecture holds in the minimal model of ADR + "θ is regular".

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 4 courses: one on the core model induction method, one on derived models associated to mice, one on hod mice and the Mouse Set Conjecture, and one shorter course on the pattern of scaled pointclasses in models of AD+.

We will meet formally 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, evenings, and weekends.

The conference organizers gratefully acknowledge financial support from the DFG (Deutsche Forschungsgemeinschaft, grant no. SCHI 484/6-1) and from the Marianne and Dr. Horst Kiesow-Stiftung, Frankfurt a.M.

Location: The talks will take place in the new building of our department (which is an annex to the old one), Orleansring 10, room no. N 2 (ground floor). Cf. below for pictures. Thanks to Andres and Grigor, more pictures are to be found here.

There will be a sequel to this conference in 2011, cf. here.

Program:

Course I. The core model induction method. (Speakers: Ralf Schindler, John Steel, Trevor Wilson)

This course will follow the book-in-progress [SchSt]. It will lay out the overall structure of core model inductions, and some of the basic core model theory and descriptive set theory they use. It will describe at least two examples, one obtaining strength from generic elementary embeddings, the second from obtaining strength from forcing axioms, via a failure of covering. The rough outline is:

(1) Hybrid mice, hybrid K. Definable iteration strategies, correct mice, and Woodin cardinals.( [SchSt, Chap. 1].)
(2) The successor stages in a core model induction. The finite stages:
(a) PD from generic embeddings.
(b) PD from forcing axioms and combinatorial principles. ([SchSt, Chap. 2].)
(3) The limit stages of core model induction in L(R):
(a) AD in L(R) from generic embeddings.
(b) AD in L(R) from forcing axioms and combinatorial principles. ([SchSt, Chaps. 3--5].)
(4) The limit stages of a core model induction in the minimal model of ADR:
(a) ADR from generic embeddings.
(b) ADR from forcing axioms and combinatorial principles. ([SchSt, Chaps. 6--7, [Sa, Chap. 5].)

Some target theorems for Course I:

(A) The existence of a precipitous ideal on ω1 plus BPFA with predicates for universally Baire sets implies PD [Cl].
(B) The existence of a homogeneous presaturated ideal on ω1 plus CH implies ADL(R) [SchSt, 2.11].
(C) The existence of an ω1-dense ideal on ω1 implies ADL(R) [SchSt].
(D) The existence of an ω1-dense ideal on ω1 plus CH plus ε implies the existence of an inner model of ADR plus ``θ is regular'' containing all the reals. [SchSt, Chap. 7], [Sa, Chap. 5].

Background iterature:

[Cl] B. Claverie, Ph.D. thesis, Münster 2010.
[Sa] Grigor Sargsyan, A tale of hybrid mice, Ph.D. thesis, Berkeley 2009.
[SchSt] R. Schindler, J. Steel, The core model induction.
[St] J. Steel, PFA implies ADL(R), J. Symb. Logic 70 (2005), pp. 1255--1296.
[StZ] J. Steel, S. Zoble, Determinacy from strong reflection.

Course II. Hod mice. (Speaker: Grigor Sargsyan)

This course will develop the theory of HOD in models of AD+ in roughly the generality in which it is now known. (This is a little past the minimal model of ADR + ``θ is regular,'' or viewed another way, a little past where HOD starts having measurable limits of Woodin cardinals.) As part of the theory, one must prove the Mouse Set Conjecture holds this far up the Wadge hierarchy. The course will be based primarily on [Sa]. The rough outline is:

(1) Computations of HODL[x,G], HODL(R), and HOD|θ0 under Mouse Capturing.
(2) Comparison theory for hod mice.
(3) The internal theory of hod mice.
(4) The construction of hod mice and the representation of HOD as a hod mouse below ADR plus θ is regular.

Some target theorems for Course II:

(A) Mouse Capturing holds up to the minimal model of ADR plus θ is regular.
(B) If there are divergent models of AD, then there is an inner model containing all the reals and satisfying ZF plus ADR plus θ is regular.
(C) Suppose that there is an active mouse with a cardinal λ which is a limit of Woodin cardinals and of cardinals which are <λ strong with respect to the predicate of being <λ strong. Then there is a pointclass Γ such that L(Γ,R) satisfies ADR plus θ is regular. In particular: Con(a Woodin limit of Woodins) implies Con(ADR plus θ is regular).

Background literature:

[Sa] Grigor Sargsyan, A tale of hybrid mice, Ph.D. thesis, Berkeley 2009.
[SchSt] R. Schindler, J. Steel, The core model induction.
[St1] J. Steel, Woodin's analysis of HODL(R).
[St2] J. Steel, An outline of inner model theory.

Course III. Derived models and mice. (Speakers: John Steel, Nam Trang)

The Derived Model Theorem is a basic expression of the correspondence between models of determinacy and models of Choice having large cardinals. In the abstract, general form of the Derived Model Theorem (due to Woodin), the Choice models are, or at least resemble closely, the HOD's of their associated determinacy models. But near the bottom of the Wadge hierarchy (and perhaps all the way), it is possible to realize the Choice models as premice. Doing this involves proving the Mouse Set Conjecture, but it goes a somewhat beyond that. It leads to some equiconistencies between large cardinal and determinacy theories.
This course will be based mainly on [St3] and [St4]. A rough outline is:

(1) Large cardinals to determinacy:
(a) The derived model below a limit λ of Woodins satisfies AD+, Σ1-reflection, and Scale(Σ21).
(b) Strong-to-λ cardinals yield Suslin representations, and hence points in the Solovay sequence of the derived model. ([L], [St 5], [St6].)
(2) Determinacy to large cardinals; every model of AD+ is a derived model:
(a) The largest Suslin cardinal case. ([St4, section 2].)
(b) Assuming ADR + (θ regular or cof(θ)=ω).([St7].)
(c) Assuming ADR + (ω < cof(θ) < θ).([St7].)
(3) Special properties of derived models of mice. The Solovay sequence in the derived model of a mouse. ([St3, sections 1--8].) AD+ plus the Mouse Set Conjecture implies every set of reals is in an R-premouse. ([St3, section 17].)
(4) Assuming AD+ plus mouse capturing plus θω1 ≤ θ, V is the derived model of a Prikry generic premouse. The consistency strength of ADR and of ADR + DC.

The main target theorem for Course III is:

ADR is equiconsistent with the ADR-hypothesis.

Background literature:

[L] P. Larson, The stationary tower, AMS Contemporary Math. series (2004).
[St3] J. Steel, Derived models associated to mice, in Computational prospects of Infinity I (Tutorials), C.T. Chong et al. eds., World Scientific (2008), pp. 105--195.
[St4] J.Steel, An optimal consistency strength lower bound for ADR, notes to be posted.
[St5] J. Steel, The derived model theorem, in Logic Colloquium 2006, Cooper et al. eds., Cambridge University Press (2009), pp. 280--327.
[St6] J. Steel, A stationary tower free proof of the derived model theorem, in Advances in Logic, Proc. of North Texas Logic Conference, Gao et al. eds., AMS Contemporary Math. series v. 425, pp. 1--7.
[St7] J. Steel, Notes on V as a derived model, handwritten notes to be posted.
[St8] J. Steel and Nam Trang, AD+, derived models, and Σ1 reflection, available here.
[YZ] Yizheng Zhu, The derived model theorem II, notes on lectures given by H. Woodin, available here.
[St9] J. Steel, Notes on the derived model theorem.

Course IV. The next scaled pointclass. (Speaker: Steve Jackson)

This shorter course will characterize the next scaled pointclass after a scaled pointclass closed under both real quantifiers, and prove other results about the pattern of Suslin cardinals and scaled pointclasses in models of AD+. The material is in [J, section 3].

Background literature:

[CKe] A. Caicedo, R. Ketchersid, A trichotomy theorem in models of AD+.
[J] S. Jackson, Structural consequences of AD, in: the Handbook of Set Theory.
[J2] S. Jackson, Slides from the course in Münster.
[J3] S. Jackson, Notes on scales.
[Ke] R. Ketchersid, More stuctural consequences of AD.

Notes taken by Martin Zeman:

part 1, part 2, part 3, part 4. (Warning: There might be typos in these notes.)

Accomodation:

Please make your reservations as soon as possible. Our conference will take place during vacation time in Nordrhein-Westfalen, and the hotels in Münster are usually booked out quite early.
  • Hotel am Schloßpark, very nice hotel, quiet and conveniently located; single rooms for 87 - 95 € per night.
  • Agora Hotel, nice hotel by the Aasee; single rooms for 70 - 75 € per night.
  • Hotel Jellentrup, quiet hotel next to the campus; single rooms for 67 - 77 € per night.
  • Jugendgästehaus am Aasee, youth hostel, but quiet and clean, by the Aasee; single rooms for 48.20 - 56.60 € per night. Book here; unfortunately, they seem to be booked out July 24-30, 2010, but they still have rooms available outside of this period of time.
  • Sleep station, budget hotel, but okay; single rooms for 32 - 40 € per night.
  • Mitwohnzentrale Münster. They arrange that you can privately rent affordable rooms/apartments from people who are travelling. This is a highly recommended option! Instruction: Go here, select "WG-Angebote" (if you would be willing to share an apartment, a "WG," with someone else) or "Wohnungsangebote" (if you'd like to have an apartment on your own) in "Münster" and click on "Suchen" (search). You'll get to a list with rooms/apartments. Please don't hesitate contacting Ralf at rds at math dot uni-muenster dot de if you need help.
  • Upon request we will try to arrange housing for you in a student apartment or student dormitory, cf. here (in German) or here (in English); prices start at about 200 € per month. They accept reservations from now on. If you are interested in this option, then you should contact Ralf at rds at math dot uni-muenster dot de as soon as possible.
  • A hotel room may also be reserved here.

    Participants:

    Dominik Adolf (Münster)
    Andrés Caicedo (Boise) July 18 -- Aug 06Germania Campus
    Sean Cox (Münster)
    Scott Cramer (Berkeley) July 16 -- Aug 31 student apartment
    Gunter Fuchs (New York)July 16 -- Aug 31student apartment
    Daisuke Ikegami (Amsterdam)July 01 -- Aug 31student apartment
    Steve Jackson (Texas)July 17 -- Aug 06Germania Campus
    Richard KetchersidJuly 18 -- Aug 06student apartment
    Paul Larson (Miami, OH)July 18 -- Aug 06Germania Campus
    Bill Mitchell (Gainesville)July 18 -- Aug 06Germania Campus
    Grigor Sargsyan (UCLA)July 15 -- Aug 06Germania Campus
    Ralf Schindler (Münster)
    Rene Schipperus (Münster)
    Philipp Schlicht (Bonn) July 16 -- Aug 31student apartment
    Farmer Schlutzenberg (Texas) July 19 -- Aug 06Sleep station
    John Steel (Berkeley) July 06 -- Aug 09Europa Haus
    Nam Trang (Berkeley)July 16 -- Aug 31student apartment
    Trevor Wilson (Berkeley)July 16 -- Aug 31student apartment
    Zhu Yizheng (Singapore)July 16 -- Aug 31student apartment
    Martin Zeman (Irvine) July 16 -- Aug 31student apartment