Forcing lightface definable well-orders without the CGH

For any given uncountable cardinal $\kappa$ with $\kappa^{{<}\kappa}=\kappa$, we present a forcing that is $<\kappa$-directed closed, has the $\kappa^+$-c.c. and introduces a lightface definable well-order of $H(\kappa^+)$. We use this to define a global iteration that does this for all such $\kappa$ simultaneously and is capable of preserving the existence of many large cardinals in the universe

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

Indestructibility of Vopenka's Principle

We show that Vopenka's Principle and Vopenka cardinals are indestructible under reverse Easton forcing iterations of increasingly directed-closed partial orders, without the need for any preparatory forcing. As a consequence, we are able to prove the relative consistency of these large cardinal axioms with a variety of statements known to be independent of ZFC, such as the generalised continuum hypothesis, the existence of a definable well-order of the universe, and the existence of morasses at many cardinals.Comment: 15 pages, submitted to Israel Journal of Mathematic

The Hurewicz dichotomy for generalized Baire spaces

By classical results of Hurewicz, Kechris and Saint-Raymond, an analytic subset of a Polish space $X$ is covered by a $K_\sigma$ subset of $X$ if and only if it does not contain a closed-in-$X$ subset homeomorphic to the Baire space ${}^\omega \omega$. We consider the analogous statement (which we call Hurewicz dichotomy) for $\Sigma^1_1$ subsets of the generalized Baire space ${}^\kappa \kappa$ for a given uncountable cardinal $\kappa$ with $\kappa=\kappa^{<\kappa}$, and show how to force it to be true in a cardinal and cofinality preserving extension of the ground model. Moreover, we show that if the Generalized Continuum Hypothesis (GCH) holds, then there is a cardinal preserving class-forcing extension in which the Hurewicz dichotomy for $\Sigma^1_1$ subsets of ${}^\kappa \kappa$ holds at all uncountable regular cardinals $\kappa$, while strongly unfoldable and supercompact cardinals are preserved. On the other hand, in the constructible universe L the dichotomy for $\Sigma^1_1$ sets fails at all uncountable regular cardinals, and the same happens in any generic extension obtained by adding a Cohen real to a model of GCH. We also discuss connections with some regularity properties, like the $\kappa$-perfect set property, the $\kappa$-Miller measurability, and the $\kappa$-Sacks measurability.Comment: 33 pages, final versio