**Contemporary Mathematics**

Volume: 690;
2017;
322 pp;
Softcover

MSC: Primary 03;
Secondary 00

**Print ISBN: 978-1-4704-2256-1
Product Code: CONM/690**

List Price: $111.00

AMS Member Price: $88.80

MAA Member Price: $99.90

**Electronic ISBN: 978-1-4704-4079-4
Product Code: CONM/690.E**

List Price: $111.00

AMS Member Price: $88.80

MAA Member Price: $99.90

# Foundations of Mathematics

Share this page *Edited by *
*Andrés Eduardo Caicedo; James Cummings; Peter Koellner; Paul B. Larson*

This volume contains the proceedings of the
Logic at Harvard conference in honor of W. Hugh Woodin's 60th
birthday, held March 27–29, 2015, at Harvard University. It
presents a collection of papers related to the work of Woodin, who has
been one of the leading figures in set theory since the early
1980s.

The topics cover many of the areas central to Woodin's work,
including large cardinals, determinacy, descriptive set theory and the
continuum problem, as well as connections between set theory and
Banach spaces, recursion theory, and philosophy, each reflecting a
period of Woodin's career. Other topics covered are forcing axioms,
inner model theory, the partition calculus, and the theory of
ultrafilters.

This volume should make a suitable introduction to Woodin's work
and the concerns which motivate it. The papers should be of interest
to graduate students and researchers in both mathematics and
philosophy of mathematics, particularly in set theory, foundations and
related areas.

#### Readership

Graduate students and research mathematicians interested in set theory, foundations, and related areas.

# Table of Contents

## Foundations of Mathematics

- Cover Cover11
- Title page v4
- Contents vii8
- Preface ix10
- Bibliography of W. Hugh Woodin xiii14
- Norming infinitesimals of large fields 122
- The enumeration degrees: Local and global structural interactions 3152
- 1. Introduction 3152
- 2. Preliminaries 3455
- 3. Step 1: An indexing of the total Δ⁰₂ degrees 3657
- 4. Step 2: An indexing of the total \IT(\zeroep) degrees 3960
- 5. Step 3: An indexing of the total Δ⁰₃ degrees 4465
- 6. Applications 4768
- 7. Determining the low enumeration degrees from the low co-d.c.e. and the co-c.e. degrees 4869
- 8. Determining the low co-d.c.e. enumeration degrees from the co-c.e. degrees 5980
- References 6687

- Ramsey properties of finite measure algebras and topological dynamics of the group of measure preserving automorphisms: Some results and an open problem 6990
- 1. Introduction 7091
- 2. Preliminaries 7293
- 3. Finite ordered measure algebras 7495
- 4. A dense subgroup of the group of measure preserving automorphisms 7899
- 5. Homogeneous measure algebras and the Ramsey Property 80101
- 6. Applications to extreme amenability and calculation of the universal minimal flow 82103
- 7. Reformulation of the Ramsey Property for homogeneous measure algebras 83104
- References 84105

- Topological Ramsey numbers and countable ordinals 87108
- 1. Introduction 88109
- 2. Preliminaries 92113
- 3. The closed pigeonhole principle for ordinals 93114
- 4. The ordinal omega plus one 96117
- 5. Stepping up by one 97118
- 6. Ordinals less than omega squared 101122
- 7. The ordinal omega squared 105126
- 8. The anti-tree partial ordering on ordinals 106127
- 9. The ordinal omega squared plus one 111132
- 10. The weak topological Erdős-Milner theorem 113134
- 11. Questions 117138
- References 118139

- Open determinacy for class games 121142
- Open problems on ultrafilters and some connections to the continuum 145166
- Obtaining Woodin’s cardinals 161182
- Woodin’s axiom (*), or Martin’s Maximum, or both? 177198
- Translation procedures in descriptive inner model theory 205226
- Implications of very large cardinals 225246
- What makes the continuum ℵ₂ 259280
- 1. Introduction 259280
- 2. The Continuum Hypothesis, the Perfect Set Property, and the Open Coloring Axiom 261282
- 3. Consequences of the Continuum Hypothesis 261282
- 4. Woodin’s Σ²₁-Absoluteness Theorem 265286
- 5. Todorcevic’s Open Coloring Axiom 265286
- 6. A Ramsey-theoretic proof that the continuum is ℵ₂ 267288
- 7. Forcing axioms and generic absoluteness 268289
- 8. Stationary set reflection and 2^{ℵ₀}≤ℵ₂ 271292
- 9. The Conjectures of Chang and Rado 273294
- 10. Woodin’s \Pmax-extension of 𝐿(\Rbb) 274295
- 11. Simply definable well orderings of \Rbb 275296
- 12. Iterated forcing and the Continuum Hypothesis 277298
- 13. The Semifilter Trichotomy 279300
- 14. Open problems 280301
- References 283304

- Set-theoretic foundations 289310
- Back Cover Back Cover1346