Combinatorial Principles in Inner Models

Martin Zeman

We shall present the Friedman-Jensen approach to the theory of fine structural inner models L[E] for overlapping extender sequences and analyse several combinatorial principles in these models. Among these principles are ♢_{κ}(A), ♢_{κ}^{+}, square_{κ}, global square, weak square principles like Jensen’s square_{κ}^{∗}, Schimmerling’s square_{κ,λ} and square_{κ,<κ}, existence of κ^{+}-Aronszajn and Souslin trees. Unlike in L, most of these principles need not hold in L[E], but we give a complete characterisation of them, that is, a condition equivalent to the failure of these principles in L[E]. This condition is in L[E] equivalent to Jensen’s weak extendibility principle and its consistency strenth ranges at the level of superstrong cardinals.

Address: Institut für Formale Logik, Univarsität Wien, Wahringer Str. 25, 1090 Wien, Austria

E-mail: zeman@mathematik.hu-berlin.de