Source: J. Symbolic Logic
Volume 74, Issue 1
We show that either of the following hypotheses imply
that there is an inner model with a proper class of strong cardinals and a proper class of Woodin cardinals. 1) There is a countably closed cardinal
κ ≥ ℵ3 such that □κ and □(κ)
fail. 2) There is a cardinal κ such that κ is weakly
compact in the generic extension by Col(κ,κ+). Of
special interest is 1) with κ = ℵ3 since it follows from PFA
by theorems of Todorcevic and Velickovic. Our main new technical result, which is due to the first author, is a weak covering theorem for the model obtained by stacking mice over Kc || κ.
A. Andretta, I. Neeman, and J. Steel, The domestic levels of $K^c$ are iterable, Israel Journal of Mathematics, vol. 125 (2001), pp. 157--201.
M. Bekkali, Topics in set theory. Lebesgue measurability, large cardinals, forcing axioms, rho-functions, Lecture Notes in Mathematics, vol. 1476, Springer-Verlag, Berlin, 1991, notes on lectures by Stevo Todorcevic.
G. Fuchs, I. Neeman, and R. Schindler, A criterion for coarse iterability, Archive for Mathematical Logic, submitted.
T. Jech, Set theory, third ed., Springer-Verlag, 2002.
R. Jensen, A new fine structure, handwritten notes, 1997, www.mathematik.hu-berlin.de/$^\sim$raesch/org/jensen.html.
--------, Robust extenders, handwritten notes, 2003, www.mathematik.hu-berlin.de/$^\sim$raesch/org/jensen.html.
R. Jensen and J. Steel in preparation.,
W. Mitchell and R. Schindler, A universal extender model without large cardinals in $V$, Journal of Symbolic Logic, vol. 69 (2004), pp. 371--386.
W. Mitchell and J. Steel, Fine structure and iteration trees, Lecture Notes in Logic, vol. 3, Springer-Verlag, 1994.
E. Schimmerling, A core model toolbox and guide, Handbook of set theory (Foreman, Kanamori, and Magidor, editors), Springer-Verlag, to appear.
--------, Combinatorial principles in the core model for one Woodin cardinal, Annals of Pure and Applied Logic, vol. 74 (1995), pp. 153--201.
--------, Coherent sequences and threads, Advances in Mathematics, vol. 216 (2007), pp. 89--117.
E. Schimmerling and J. Steel, The maximality of the core model, Transactions of the American Mathematical Society, vol. 351 (1999), pp. 3119--3141.
E. Schimmerling and M. Zeman, Characterization of $\square_\kappa$ in core models, Journal of Mathematical Logic, vol. 4 (2004), pp. 1--72.
R. Schindler and J. Steel, The core model induction, book in preparation.
R. Schindler and M. Zeman, Fine structure theory, Handbook of set theory (Foreman, Kanamori, and Magidor, editors), Springer-Verlag, to appear.
J. Steel, The derived model theorem, preprint, math.berkeley.edu/$^\sim$steel/papers/Publications.html.
--------, An outline of inner model theory, Handbook of set theory (Foreman, Kanamori, and Magidor, editors), Springer-Verlag, to appear.
--------, The core model iterability problem, Lecture Notes in Logic, vol. 8, Springer-Verlag, 1996.
--------, \sf PFA implies $\sf AD^L(\mathbb R)$, Journal of Symbolic Logic, vol. 70 (2005), pp. 1255--1296.
S. Todorcevic, A note on the Proper Forcing Axiom, Contemporary Mathematics, vol. 95 (1984), pp. 209--218.
Mathematical Reviews (MathSciNet): MR763902
B. Velickovic, Forcing axioms and stationary sets, Advances in Mathematics, vol. 94 (1992), pp. 256--284.
M. Zeman, Inner models and large cardinals, de Gruyter, Berlin, New York, 2002.