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

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 . Thanks to Andres and Grigor, more are to be found here.

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

(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 AD

(a) AD

(b) AD

Some target theorems for Course I:

(A) The existence of a precipitous ideal on ω

(B) The existence of a homogeneous presaturated ideal on ω

(C) The existence of an ω

(D) The existence of an ω

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 AD

[StZ] J. Steel, S. Zoble, Determinacy from strong reflection.

(1) Computations of HOD

(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 AD

Some target theorems for Course II:

(A) Mouse Capturing holds up to the minimal model of AD

(B) If there are divergent models of AD, then there is an inner model containing all the reals and satisfying ZF plus AD

(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 AD

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 HOD

[St2] J. Steel, An outline of inner model theory.

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

(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

(a) The largest Suslin cardinal case. ([St4, section 2].)

(b) Assuming AD

(c) Assuming AD

(3) Special properties of derived models of mice. The Solovay sequence in the derived model of a mouse. ([St3, sections 1--8].) AD

(4) Assuming AD

The main target theorem for Course III is:

AD

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 AD

[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

[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.

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.

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

Dominik Adolf (Münster) | ||

Andrés Caicedo (Boise) | July 18 -- Aug 06 | Germania Campus |

Sean Cox (Münster) | ||

Scott Cramer (Berkeley) | July 16 -- Aug 31 | student apartment |

Gunter Fuchs (New York) | July 16 -- Aug 31 | student apartment |

Daisuke Ikegami (Amsterdam) | July 01 -- Aug 31 | student apartment |

Steve Jackson (Texas) | July 17 -- Aug 06 | Germania Campus |

Richard Ketchersid | July 18 -- Aug 06 | student apartment |

Paul Larson (Miami, OH) | July 18 -- Aug 06 | Germania Campus |

Bill Mitchell (Gainesville) | July 18 -- Aug 06 | Germania Campus |

Grigor Sargsyan (UCLA) | July 15 -- Aug 06 | Germania Campus |

Ralf Schindler (Münster) | ||

Rene Schipperus (Münster) | ||

Philipp Schlicht (Bonn) | July 16 -- Aug 31 | student apartment |

Farmer Schlutzenberg (Texas) | July 19 -- Aug 06 | Sleep station |

John Steel (Berkeley) | July 06 -- Aug 09 | Europa Haus |

Nam Trang (Berkeley) | July 16 -- Aug 31 | student apartment |

Trevor Wilson (Berkeley) | July 16 -- Aug 31 | student apartment |

Zhu Yizheng (Singapore) | July 16 -- Aug 31 | student apartment |

Martin Zeman (Irvine) | July 16 -- Aug 31 | student apartment |