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

Eastonʼs theorem and large cardinals from the optimal hypothesis

AbstractThe equiconsistency of a measurable cardinal with Mitchell order o(κ)=κ++ with a measurable cardinal such that 2κ=κ++ follows from the results by W. Mitchell (1984) [13] and M. Gitik (1989) [7]. These results were later generalized to measurable cardinals with 2κ larger than κ++ (see Gitik, 1993 [8]).In Friedman and Honzik (2008) [5], we formulated and proved Eastonʼs (1970) theorem [4] in a large cardinal setting, using slightly stronger hypotheses than the lower bounds identified by Mitchell and Gitik (we used the assumption that the relevant target model contains H(μ), for a suitable μ, instead of the cardinals with the appropriate Mitchell order).In this paper, we use a new idea which allows us to carry out the constructions in Friedman and Honzik (2008) [5] from the optimal hypotheses. It follows that the lower bounds identified by Mitchell and Gitik are optimal also with regard to the general behavior of the continuum function on regulars in the context of measurable cardinals

Halmazelméleti topológia = Set-theoretic topology

Ebben az OTKA-pályázatban -- kutatási tervünknek megfelelően -- kutatásokat végeztünk és jelentős eredményeket értünk el a következő négy területen: (I) Halmazelméleti topológia (kompakt terek, szétszórt terek, számosságfüggvények, felbonthatóság) (II) Leíró halmazelmélet (III) Végtelen és véges kombinatorika (IV) Valós analízis és mértékelmélet . Eredményeinket 45 dolgozatban írtuk le, amelyek túlnyomó többsége a megfelelő terület legrangosabb nemzetközi folyóirataiban jelentek meg, illetve fognak megjelenni (ezek közül 6 dolgozatot már benyujtottunk, de eddig még nem lettek elfogadva). Kutatócsoportunk 8 résztvevővel indult, de sajnos egyikünk -- Gerlits János) 2008-bqn elhunyt. Kutatási eredményeinkről számos nemzetközi konferencián is számot adtunk, sok esetben közülünk négyen (Elekes, Juhász, Mátrai, Soukup) mint plenáris és/vagy meghívott előadó. | In the present project, following our research plan, we have done research and established a number of significant results in the following four areas: (I) Set-theoretic topology (compact spaces, scattered spaces, cardinal functions, resolvability) (II) Descriptive set-theory (III) Infinite and finite combinatorics (IV) Real analysis and measure theory We presented our results in 45 papers almost all of which appeared or will appear in the leading international journals of these fields (6 of these papers have been submitted but not accepted as yet). Our research group consisted of 8 people, one of us -- J. Gerlits -- unfortunately passed away in 2008. We also participated at a large number of international conferences, four of us (Elekes, Juhász, Mátrai, Soukup) as plenary and/or invited speakers at many of these