Mini-Workshop: Feinstrukturtheorie und Innere Modelle

The main aim of fine structure theory and inner model theory can be summarized as the construction of models which have a canonical inner structure (a fine structure), making it possible to analyze them in great detail, and which at the same time reflect important aspects of the surrounding mathematical universe, in that they satisfy certain strong axioms of infinity, or contain complicated sets of reals. Applications range from obtaining lower bounds on the consistency strength of all sorts of set theoretic principles in terms of large cardinals, to proving the consistency of certain combinatorial properties, their compatibility with strong axioms of infinity, or outright proving results in descriptive set theory (for which no proofs avoiding fine structure and inner models are in sight). Fine structure theory and inner model theory has become a sophisticated and powerful apparatus which yields results that are among the deepest in set theory

Set-Theoretic Geology

A ground of the universe V is a transitive proper class W subset V, such that W is a model of ZFC and V is obtained by set forcing over W, so that V = W[G] for some W-generic filter G subset P in 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.Comment: 44 pages; commentary concerning this article can be made at http://jdh.hamkins.org/set-theoreticgeology

On Recurrence Axioms

The Recurrence Axiom for a class $\mathcal{P}$ of \pos\ and a set $A$ of parameters is an axiom scheme in the language of ZFC asserting that if a statement with parameters from $A$ is forced by a poset in $\mathcal{P}$, then there is a ground containing the parameters and satisfying the statement. The tightly super-$C^{(\infty)}$-$\mathcal{P}$-Laver generic hyperhuge continuum implies the Recurrence Axiom for $\mathcal{P}$ and $\mathcal{H}(2^{\aleph_0})$. The consistency strength of this assumption can be decided thanks to our main theorems asserting that the minimal ground (bedrock) exists under a tightly $\mathcal{P}$-generic hyperhuge cardinal $\kappa$, and that $\kappa$ in the bedrock is genuinely hyperhuge, or even super $C^{(\infty)}$ hyperhuge if $\kappa$ is a tightly super-$C^{(\infty)}$-$\mathcal{P}$-Laver generic hyperhuge definable cardinal. The Laver Generic Maximum (LGM), one of the strongest combinations of axioms in our context, integrates practically all known set-theoretic principles and axioms in itself, either as its consequences or as theorems holding in (many) grounds of the universe. For example, double plus version of Martin's Maximum is a consequence of LGM while Cicho\'n's Maximum is a phenomenon in many grounds of the universe under LGM

Inner models with large cardinal features usually obtained by forcing

We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal \kappa for which 2^\kappa=\kappa^+, another for which 2^\kappa=\kappa^++ and another in which the least strongly compact cardinal is supercompact. If there is a strongly compact cardinal, then there is an inner model with a strongly compact cardinal, for which the measurable cardinals are bounded below it and another inner model W with a strongly compact cardinal \kappa, such that H_{\kappa^+}^V\subseteq HOD^W. Similar facts hold for supercompact, measurable and strongly Ramsey cardinals. If a cardinal is supercompact up to a weakly iterable cardinal, then there is an inner model of the Proper Forcing Axiom and another inner model with a supercompact cardinal in which GCH+V=HOD holds. Under the same hypothesis, there is an inner model with level by level equivalence between strong compactness and supercompactness, and indeed, another in which there is level by level inequivalence between strong compactness and supercompactness. If a cardinal is strongly compact up to a weakly iterable cardinal, then there is an inner model in which the least measurable cardinal is strongly compact. If there is a weakly iterable limit \delta of <\delta-supercompact cardinals, then there is an inner model with a proper class of Laver-indestructible supercompact cardinals. We describe three general proof methods, which can be used to prove many similar results