A tale of hybrid mice

We develop the basic theory of rigidly layered hod mice below ADR + “Θ is regular”. Woven into this theory is a proof of Mouse Set Conjecture. Once the basic theory is available, it can be applied to questions of evaluating consistency strengths of set theoretic hypothesis in terms of large cardinals. One such application is that the consistency of ADR + “Θ is regular” follows from the consistency of a Woodin limit of Woodins. Other applications use the core model induction. Here, we consider the problem of obtaining lower bounds of consistency strengths for the theory CH + “ there is an ω1-dense ideal on ω1”. We show that under one additional assumption on the generic embedding given by the ideal, namely that its restriction to ordinals is independent of the generic object, ADR + “Θ is regular” is a lower bound. Woodin showed the consistency of this hypothesis relative to ADR + “Θ is regular”. Combining the two results, we get an equiconsistency.

[1]  Ralf Schindler,et al.  The self-iterability of L[E] , 2009, J. Symb. Log..

[2]  Yiannis N. Moschovakis.  Ordinal games and playful models , 1981 .

[3]  Ralf Schindler,et al.  Stacking mice , 2009, The Journal of Symbolic Logic.

[4]  Y. Moschovakis.  Descriptive Set Theory , 1980 .

[5]  Robert van Wesep,et al.  Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II: Wadge degrees and descriptive set theory , 1978 .

[6]  Itay Neeman.  Optimal proofs of determinacy , 1995, Bull. Symb. Log..

[7]  Ralf Schindler,et al.  The core model induction , 2007 .

[8]  James Cummings,et al.  The Tree Property , 1998 .

[9]  Ralf Schindler,et al.  Deconstructing Inner Model Theory , 2002, J. Symb. Log..

[10]  John R. Steel.  Inner Models with Many Woodin Cardinals , 1993, Ann. Pure Appl. Logic.

[11]  William J. Mitchell,et al.  A universal extender model without large cardinals in V , 2004, J. Symb. Log..

[12]  P. Welch,et al.  THE AXIOM OF DETERMINACY, FORCING AXIOMS, AND THE NONSTATIONARY IDEAL (de Gruyter Series in Logic and its Applications 1) , 2001 .

[13]  Yiannis N. Moschovakis,et al.  The axiom of determinacy, strong partition properties and nonsingular measures , 1981 .

[14]  Alexander S. Kechris,et al.  The theory of countable analytical sets , 1975 .

[15]  A. Kechris.  Measure and category in effective descriptive set theory , 1973 .

[16]  John R. Steel.  Scales in K(R) , 2008 .

[17]  W. Hugh Woodin,et al.  The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal , 1999 .

[18]  J. R. Steel.  Logic Colloquium 2006: The derived model theorem , 2009 .

[19]  John R. Steel.  Scales in K(R) at the end of a weak gap , 2008, J. Symb. Log..

[20]  Itay Neeman.  Inner models in the region of a Woodin limit of Woodin cardinals , 2002, Ann. Pure Appl. Log..

[21]  A. Kanamori.  The Higher Infinite , 1994 .

[22]  Martin Zeman.  Inner Models and Large Cardinals , 2001 .

[23]  John R. Steel.  PFA implies ADL(ℝ) , 2005, J. Symb. Log..

[24]  J. Steel.  Derived models associated to mice , 2007 .

[25]  Alexander S. Kechris,et al.  Sets of ordinals constructible from trees and the third Victoria Delfino problem , 1984 .

[26]  Ralf Schindler,et al.  The strength of choiceless patterns of singular and weakly compact cardinals , 2009, Ann. Pure Appl. Log..

[27]  M. Foreman.  Ideals and Generic Elementary Embeddings , 2010 .

[28]  R. Solovay,et al.  Introduction to Q-Theory , 1983 .

[29]  Ernest Schimmerling.  The ABC's of mice , 2001, Bull. Symb. Log..

[30]  Leo Harrington,et al.  On the determinacy of games on ordinals , 1981 .

[31]  John R. Steel.  Projectively Well-Ordered Inner Models , 1995, Ann. Pure Appl. Log..