Fine structure from normal iterability

Farmer Schlutzenberg

Sonstige wissenschaftliche Veröffentlichung

Zusammenfassung

We show that (i) the standard fine structural properties for premice follow from normal iterability (whereas the classical proof relies on iterability for stacks of normal trees), and (ii) every mouse which is finitely generated above its projectum, is an iterate of its core.That is, let m be an integer and let M be an m-sound, (m,ω_1+1)-iterable premouse. Then (i) M is (m+1)-solid and (m+1)-universal, (m+1)condensation holds for M, and if m≥1 then M is super-Dodd-sound, a slight strengthening of Dodd-soundness. And (ii) if there is x ∈ M such that M is the rΣm+1-hull of parameters in (ρ^M_(m+1))∪{x}, then M is a normal iterate of its (m+1)-core C = ℭ_(m+1)(M); in fact, there is an m-maximal iteration tree T on C, of finite length, such that M = M^T_∞, and i^T_∞ is just the core embedding. Applying fact (ii), we prove that if M ⊨ ZFC is a mouse and W ⊆ M is a ground of M via a strategically σ-closed forcing ℙ ∈ W, and if M | ℵ^M_1 ∈ W (that is, the initial segment of M of height ℵ^M_1 is in W), then the forcing is trivial; that is, M ⊆ W. And if there is a measurable cardinal, then there is a non-solid premouse. The results hold for premice with Mitchell-Steel indexing, allowing extenders of superstrong type to appear on the extender sequence.