W. Hugh Woodin

The axiom of determinacy, forcing axioms, and the nonstationary ideal

This is the revised edition of a well-established monograph on the identification of a canonical model in which the Continuum Hypothesis is false. Written by an expert in the field, it is directed to researchers and advanced graduate students in Mathematical Logic and Set Theory. The second edition is updated to take into account some of the developments in the decade since the first edition appeared, this includes a revised discussion of ?-logic and related matters.

Large Cardinals from Determinacy

This chapter gives an account of Woodin’s general technique for deriving large cardinal strength from determinacy hypotheses. These results appear here for the first time and the treatment is self-contained.

SUITABLE EXTENDER MODELS II: BEYOND ω-HUGE

We investigate large cardinal axioms beyond the level of ω-huge in context of the universality of the suitable extender models of [Suitable Extender Models I, J. Math. Log.10 (2010) 101–339]. We show that there is an analog of ADℝ at the level of ω-huge, more precisely the construction of the minimum model of ADℝ generalizes to the level of Vλ+1. This allows us to formulate the indicated generalization of ADℝ and then to prove that if the axiom holds in V at a proper class of λ then in every suitable extender model, the axiom holds at a proper class of λ (provided the relevant ω-huge embeddings can be chosen to preserve the suitable extender model).

THE GROUND AXIOM IS CONSISTENT WITH V ≠ HOD

The Ground Axiom asserts that the universe is not a nontrivial set-forcing extension of any inner model. Despite the apparent second-order nature of this assertion, it is first-order expressible in set theory. The previously known models of the Ground Axiom all satisfy strong forms of V = HOD. In this article, we show that the Ground Axiom is relatively consistent with V ≠ HOD. In fact, every model of ZFC has a class-forcing extension that is a model of ZFC + GA + V ≠ HOD. The method accommodates large cardinals: every model of ZFC with a supercompact cardinal, for example, has a class-forcing extension with ZFC + GA + V ≠ HOD in which this supercompact cardinal is preserved.

The Necessary Maximality Principle for c. c. c. forcing is equiconsistent with a weakly compact cardinal

The Necessary Maximality Principle for c.c.c. forcing with real parameters is equiconsistent with the existence of a weakly compact cardinal. The Necessary Maximality Principle for c.c.c. forcing, denoted 2mpccc(R), asserts that any statement about a real in a c.c.c. extension that could be- come true in a further c.c.c. extension and remain true in all subsequent c.c.c. extensions, is already true in the minimal extension containing the real. We show that this principle is equiconsistent with the existence of a weakly compact cardinal. The principle is one of a family of principles considered in (Ham03) (build- ing on ideas of (Cha00) and overlapping with independent work in (SV01)). MSC: 03E55, 03E40. Keywords: forcing axiom, ccc forcing, weakly compact cardinal. The first author is affiliated with the College of Staten Islandof CUNY and The CUNY Graduate Center, and his research has been supported by grants from the Research Foun- dation of CUNY and the National Science Foundation DMS-9970993. The research of the second author has been partially supported by National Science Foundation grant DMS-9970255.

Generic codes for uncountable ordinals, partition properties, and elementary embeddings

This paper was circulated in handwritten form in December 1980 and contained Sections 1-7 below. There are two additionalSections 8 and 9 here that contain further material and comments.

Extending partial orders to dense linear orders

Abstract J. Łoś raised the following question: Under what conditions can a countable partially ordered set be extended to a dense linear order merely by adding instances of comparability (without adding new points)? We show that having such an extension is a Σ 1 l -complete property and so there is no Borel answer to Łośs question. Additionally, we show that there is a natural Π 1 l -norm on the partial orders which cannot be so extended and calculate some natural ranks in that norm.

The cardinals below |[ω1]<ω1|

Abstract The results of this paper concern the effective cardinal structure of the subsets of [ ω 1 ] ω 1 , the set of all countable subsets of ω 1 . The main results include dichotomy theorems and theorems which show that the effective cardinal structure is complicated.

Forcing, Iterated Ultrapowers, and Turing Degrees

This volume presents the lecture notes of short courses given by three leading experts in mathematical logic at the 2010 and 2011 Asian Initiative for Infinity Logic Summer Schools. The major topics covered set theory and recursion theory, with particular emphasis on forcing, inner model theory and Turing degrees, offering a wide overview of ideas and techniques introduced in contemporary research in the field of mathematical logic.

Notes on forcing axioms

The Baire Category Theorem and the Baire Category Numbers Coding into the Reals Descriptive Set-Theoretic Consequences Measure-Theoretic Consequences Variations on the Souslin Hypothesis The S- and L-Space Problems The Side-Condition Method Ideal Dichotomies Coherent and Lipschitz Trees Applications to the S-Space Problem and the Von Neumann Problem Biorthogonal Systems Structure of Compact Spaces Ramsey Theory on Ordinals Five Cofinal Types Five Linear Orderings mm and Cardinal Arithmetic Reflection Principles Appendices: Basic Notions Preserving Stationary Sets Historical and Other Comments.