CONTENTS -- Foreword -- Preface -- Section I. Invited Lectures -- Absoluteness, Truth, and Quotients -- 1. Finitism, 'Countablism' and a Little Bit Further -- 2. Independence -- 2.1. The story of projective sets -- 3. Absoluteness -- 3.1. Beyond projective sets -- 3.2. Absoluteness and the uncountable -- 3.3. Level by level -- 4. Third-Order Arithmetic -- 4.1. Conditional absoluteness -- 4.2. Π21? -- 5. Quotient Borel Structures -- 5.1. Trivial automorphisms -- 5.2. C*-algebras and their multipliers -- 5.3. General rigidity conjectures -- Acknowledgments -- Appendix -- A.1. Hereditary sets -- A.2. Arithmetical formulas -- A.3. Analytical formulas -- A.4. Examples -- References -- A Multiverse Perspective on the Axiom of Constructibility -- 1. Introduction -- 2. Some New Problems with Maddy's Proposal -- 3. Several Ways in which V = L is Compatible with Strength -- 4. An Upwardly Extensible Concept of Set -- References -- Hilbert, Bourbaki and the Scorning of Logic -- 1917: Hilbert returns to the foundations of mathematics -- Some terminology -- 1928: publication of the treatise of Hilbert and Ackermann -- Logic in the twenties -- 1922: the Hilbert operator is launched -- 1928: Hilbert at Bologna -- 1928: the war of the Frogs and the Mice -- 1929: the completeness theorem -- 1930/31: the incompleteness theorems -- 1931/34: Hilbert's delayed response to the incompleteness theorems -- Hilbert's programme after Godel -- 1934, 1939: publication in two volumes of the treatise of Hilbert and Bernays -- 1935: the naissance of Bourbaki -- Bourbaki's syntax -- The length of τ-expansions -- Every null term is equal to a proper term -- Any two null terms are equal -- Perverted interpretation of quantifiers -- Discussion -- Bourbaki's remarks on progress in logic -- Bourbaki's account of the incompleteness theorem -- Godement's formal system.

Godement's set-theoretic axioms -- Misunderstandings of work of logicians -- Unease in the presence of logic -- page 7: the problem of choice -- Discussion of equality -- A list of axioms of set theory -- The legacy of Napoleon: the foundation of the modern French university system -- Politics and mathematics -- Bourbaki and French nationalism -- The chimera of completeness -- La Tribu -- Bourbaki consult Rosser -- Why use Bourbaki's formalisation? -- Structuralism: a part but not the whole of mathematics -- Another collapse -- Back to St Benedict -- Acknowledgments -- References -- Toward Objectivity in Mathematics -- 1. Objectivity and Objectivism -- 2. Mathematics as Part of Human Knowledge -- 3. Set Theory and the Unity of Mathematics -- 4. Set-Theoretic Realism -- 4.1. An epistemological question -- 4.2. The intrinsicist answer -- 4.3. The "testable consequences" answer -- 4.4. The Thin Realist answer -- 5. Insights from Reverse Mathematics -- 6. Wider Cultural Significance? -- Acknowledgment -- References -- Sort Logic and Foundations of Mathematics -- 1. Introduction -- 2. Sort Logic -- 2.1. Basic concepts -- 2.2. Syntax -- 2.3. Axioms -- 2.4. Semantics -- 3. Sort Logic and Set Theory -- 4. Sort Logic and Foundations of Mathematics -- References -- Reasoning about Constructive Concepts -- 1. -- 2. -- 3. -- 4. -- 5. -- 6. -- References -- Perfect Infinities and Finite Approximation -- 1. Introduction -- 2. Continuity and its Alternatives -- 2.1. -- 2.2. -- 2.3. -- 2.4. -- 3. In Search of Logically Perfect Structures -- 3.1. -- 3.2. -- 3.3. -- 3.4. Topological structures -- 3.5. -- 3.6. -- 3.7. -- 3.8. -- 4. Structural Approximation -- 4.1. Topological structures -- 4.2. Structural approximation -- 5. Examples -- 5.1. Metric spaces -- 5.2. Cyclic groups in profinite topology -- 5.3. The ring of p-adic integers -- 5.4. Compactified groups.

5.5. Cyclic groups in metric topology and their compactifications -- 5.6. 2-ends compactification of Z -- 6. Approximation by Some Finite Structures -- 6.1. Approximation by finite fields -- 6.2. Approximation by finite groups -- References -- Section II. Special Session -- An Objective Justification for Actual Infinity? -- 1. Introduction -- 2. Objectivity in Mathematics -- 3. Potential Infinity versus Actual Infinity -- 4. Insights from Reverse Mathematics -- References -- Oracle Questions -- 1. Introduction -- 2. Questions -- 2.1. Ilijas Farah -- 2.2. Moti Gitik -- 2.3. Joel David Hamkins -- 2.4. Juliette Kennedy -- 2.5. Steffen Lempp -- 2.6. Stephen G. Simpson -- 2.7. Theodore Slaman -- 2.8. Jouko Vaananen -- 2.9. Nik Weaver -- 2.10. W. Hugh Woodin -- 2.11. Boris Zilber -- References.