Preserving levels of projective determinacy by tree forcings

We prove that various classical tree forcings -- for instance Sacks forcing, Mathias forcing, Laver forcing, Miller forcing and Silver forcing -- preserve the statement that every real has a sharp and hence analytic determinacy. We then lift this result via methods of inner model theory to obtain level-by-level preservation of projective determinacy (PD). Assuming PD, we further prove that projective generic absoluteness holds and no new equivalence classes classes are added to thin projective transitive relations by these forcings.Comment: 3 figure

Mathias and silver forcing parametrized by density

We define and investigate versions of Silver and Mathias forcing with respect to lower and upper density. We focus on properness, Axiom A, chain conditions, preservation of cardinals and adding Cohen reals. We find rough forcings that collapse 2ω to ω , while others are surprisingly gentle. We also study connections between regularity properties induced by these parametrized forcing notions and the Baire property

Cardinal invariants of the continuum -A survey

Abstract These are expanded notes of a series of two lectures given at the meeting on axiomatic set theory at Kyōto University in November 2000. The lectures were intended to survey the state of the art of the theory of cardinal invariants of the continuum, and focused on the interplay between iterated forcing theory and cardinal invariants, as well as on important open problems. To round off the present written account of this survey, we also include sections on ZF C-inequalities between cardinal invariants, and on applications outside of set theory. However, due to the sheer size of the area, proofs had to be mostly left out. While being more comprehensive than the original talks, the personal flavor of the latter is preserved in the notes. Some of the material included was presented in talks at other conferences

Alternative Cichoń Diagrams and Forcing Axioms Compatible with CH

This dissertation surveys several topics in the general areas of iterated forcing, infinite combinatorics and set theory of the reals. There are two parts. In the first half I consider alternative versions of the Cichoń diagram. First I show that for a wide variety of reduction concepts there is a Cichoń diagram for effective cardinal characteristics relativized to that reduction. As an application I investigate in detail the Cichoń diagram for degrees of constructibility relative to a fixed inner model of ZFC. Then I study generalizations of cardinal characteristics to the space of functions from Baire space to Baire space. I prove that these cardinals can be organized into two diagrams analogous to the standard Cichoń diagram show several independence results and investigate their relation to cardinal invariants on omega. In the second half of the thesis I look at forcing axioms compatible with CH. First I consider Jensen\u27s subcomplete and subproper forcing. I generalize these notions to larger classes which are (apparently) much more nicely behaved structurally. I prove iteration and preservation theorems for both classes and use these to produce many new models of the subcomplete forcing axiom. Finally I deal with dee-complete forcing and its associated axiom DCFA. Extending a well-known result of Shelah, I show that if a tree of height omega one with no branch can be embedded into an omega one tree, possibly with uncountable branches, then it can be specialized without adding reals. As a consequence I show that DCFA implies there are no Kurepa trees, even if CH fails

Kolmogorov complexity and computably enumerable sets

We study the computably enumerable sets in terms of the: (a) Kolmogorov complexity of their initial segments; (b) Kolmogorov complexity of finite programs when they are used as oracles. We present an extended discussion of the existing research on this topic, along with recent developments and open problems. Besides this survey, our main original result is the following characterization of the computably enumerable sets with trivial initial segment prefix-free complexity. A computably enumerable set $A$ is $K$-trivial if and only if the family of sets with complexity bounded by the complexity of $A$ is uniformly computable from the halting problem