Gunter Fuchs

SET-THEORETIC GEOLOGY

Abstract A ground of the universe V is a transitive proper class W ⊆ V , such that W ⊨ ZFC and V is obtained by set forcing over W, so that V = W [ G ] for some W-generic filter G ⊆ P ∈ W . The model V satisfies the ground axiom GA if there are no such W properly contained in V. The model W is a bedrock of V if W is a ground of V and satisfies the ground axiom. The mantle of V is the intersection of all grounds of V. The generic mantle of V is the intersection of all grounds of all set-forcing extensions of V. The generic HOD, written gHOD, is the intersection of all HODs of all set-forcing extensions. The generic HOD is always a model of ZFC, and the generic mantle is always a model of ZF. Every model of ZFC is the mantle and generic mantle of another model of ZFC. We prove this theorem while also controlling the HOD of the final model, as well as the generic HOD. Iteratively taking the mantle penetrates down through the inner mantles to what we call the outer core, what remains when all outer layers of forcing have been stripped away. Many fundamental questions remain open.

HIERARCHIES OF FORCING AXIOMS, THE CONTINUUM HYPOTHESIS AND SQUARE PRINCIPLES

I analyze the hierarchies of the bounded and the weak bounded forcing axioms, with a focus on their versions for the class of subcomplete forcings, in terms of implications and consistency strengths. For the weak hierarchy, I provide level-by-level equiconsistencies with an appropriate hierarchy of partially remarkable cardinals. I also show that the subcomplete forcing axiom implies Larson’s ordinal reflection principle at ω 2 , and that its effect on the failure of weak squares is very similar to that of Martin’s Maximum.

λ-structures and s-structures: Translating the models

Abstract I develop a translation procedure between λ -structures, which correspond to premice in the Friedman–Jensen indexing convention on the one hand and s -structures, which are essentially the same as premice in the Mitchell–Steel indexing scheme.

ON SEQUENCES GENERIC IN THE SENSE OF MAGIDOR

The main result of this paper is a combinatorial characterization of Magidor-generic sequences. Using this characterization, I show that the critical sequences of certain iterations are Magidor-generic over the target model. I then employ these results in order to analyze which other Magidor sequences exist in a Magidor extension. One result in this direction is that if we temporarily identify Magidor sequences with their ranges, then Magidor sequences are maximal, in the sense that they contain any other Magidor sequence that is present in their forcing extension, even if the other sequence is generic for a different Magidor forcing. A stronger result holds if both sequences come from the same forcing: I show that a Magidor sequence is almost unique in its forcing extension, in the sense that any other sequence generic for the same forcing which is present in the same forcing extension coincides with the original sequence at all but finitely many coordinates, and at all limit coordinates. Further, I ask the question: If d ∈ V[c], where c and d are Magidor-generic over V, then which Magidor forcing can d be generic for? It turns out that it must essentially be a collapsed version of the Magidor forcing for which c was generic. I treat several related questions as well. Finally, I introduce a special case of Magidor forcing which I call minimal Magidor forcing. This approach simplifies the forcing, and I prove that it doesn’t restrict the class of possible Magidor sequences. I.e., if c is generic for a Magidor forcing over V, then it is generic for a minimal Magidor forcing over V.

A criterion for coarse iterability

The main result of this paper is the following theorem: Let M be a premouse with a top extender, F. Suppose that (a) M is linearly coarsely iterable via hitting F and its images, and (b) if M* is a linear iterate of M as in (a), then M* is coarsely iterable with respect to iteration trees which do not use the top extender of M* and its images. Then M is coarsely iterable.

The subcompleteness of Magidor forcing

It is shown that the Magidor forcing to collapse the cofinality of a measurable cardinal that carries a length

Closure properties of parametric subcompleteness

\omega _1

Successor levels of the Jensen hierarchy

ω1 sequence of normal ultrafilters, increasing in the Mitchell order, to