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Book Overview
Table of Contents
Chapter Abstracts
Resources
On Nietzsche
The Logic of Metaphor
The Singularity

Preface
Sets
 1. Collections of Things
 2. Sets and Members
 3. Set Builder Notation
 4. Subsets
 5. Small Sets
 6. Unions of Sets
 7. Intersections of Sets
 8. Difference of Sets
 9. Set Algebra
 10. Sets of Sets
 11. Union of a Set of Sets
 12. Power Sets
 13. Sets and Selections
 14. Pure Sets
 15. Sets and Numbers
 16. Sums of Sets of Numbers
 17. Ordered Pairs
 18. Ordered Tuples
 19. Cartesian Products
Relations
 1. Relations
 2. Some Features of Relations
 3. Equivalence Relations and Classes
 4. Closures of Relations
 5. Recursive Definitions and Ancestrals
 6. Personal Persistence
 6.1 The Diachronic Sameness Relation
 6.2 The Memory Relation
 6.3 Symmetric then Transitive Closure
 6.4 The Fission Problem
 6.5 Transitive then Symmetric Closure
 7. Closure Under an Operation
 8. Closure Under Physical Relations
 9. Order Relations
 10. Degrees of Perfection
 11. Parts of Sets
 12. Functions
 13. Some Examples of Functions
 14. Isomorphisms
 15. Functions and Sums
 16. Sequences and Operations on Sequences
 17. Cardinality
 18. Sets and Classes
Machines
 1. Machines
 2. Finite State Machines
 2.1 Rules for Machines
 2.2 The Careers of Machines
 2.3 Utilities of States and Careers
 3. Networks of Machines
 3.1 Interacting Machines
 3.2 Machines in the Network
 3.3 Configurations in the Network
 3.4 The History of a Network of Machines
 3.5 Mechanical Metaphysics
 4. The Game of Life
 4.1 A Universe Made from Machines
 4.2 The Causal Law in the Game of Life
 4.3 Regularities in the Causal Flow
 5. Turing Machines
Semantics
 1. Extensional Semantics
 1.1 Words and Referents
 1.2 A Sample Vocabulary and Model
 1.3 Sentences and TruthConditions
 2. Simple Modal Semantics
 2.1 Possible Worlds
 2.2. A Sample Modal Structure
 2.3 Sentences and Truth at Possible Worlds
 2.4 Modalities
 2.5 Intensions
 2.6 Propositions
 3. Modal Semantics with Counterparts
 3.1 The Counterpart Relation
 3.2 A Sample Model for Counterpart Theoretic Semantics
 3.3 TruthConditions for NonModal Statements
 3.4 TruthConditions for Modal Statements
Probability
 1. Sample Spaces
 2. Simple Probability
 3. Combined Probabilities
 4. Probability Distributions
 5. Conditional Probabilities
 5.1 Restricting the Sample Space
 5.2 The Definition of Conditional Probability
 5.3 An Example Involving Marbles
 5.4 Independent Events
 6. Bayes Theorem
 6.1 The First Form of Bayes Theorem
 6.2 An Example Involving Medical Diagnosis
 6.3 The Second Form of Bayes Theorem
 6.4 An Example Involving Envelopes with Prizes
 7. Degrees of Belief
 7.1 Sets and Sentences
 7.2 Subjective Probability Functions
 8. Bayesian Confirmation Theory
 8.1 Confirmation and Disconfirmation
 8.2 Bayesian Conditionalization
 9. Knowledge and the Flow of Information
Utilitarianism
 1. Act Utilitarianism
 1.1 Agents and Actions
 1.2 Actions and their Consequences
 1.3 Utility and Moral Quality
 2. Expected Utility
 3. World Utilitarianism
 3.1 Agents and their Careers
 3.2 Careers and Compatible Worlds
 3.3 Runs and Prefixes
 3.4 Runs and the Worlds Open to Them
 3.5 The Utilities of Worlds
 3.6 Optimal Open Worlds
 3.7 Actions and their Moral Qualities
From the Finite to the Infinite
 1. Recursively Defined Series
 2. Limits of Recursively Defined Series
 2.1 Counting Through All the Numbers
 2.2 Cantor's Three Number Generating Rules
 2.3 The Series of Von Neumann Numbers
 3. Some Examples of Series with Limits
 3.1 Achilles Runs on Zeno's Racetrack
 3.2 The Royce Map
 3.3 The Hilbert Paper
 3.4 An Endless Series of Degrees of Perfection
 4. Infinity
 4.1 Infinity and Infinite Complexity
 4.2 The Hilbert Hotel
 4.3 Operations on Infinite Sequences
 5. Supertasks
 5.1 Reading the Borges Book
 5.2 The Thomson Lamp
 5.3 Zeus Performs a SuperComputation
 5.4 Accelerating Turing Machines
Bigger Infinities
 1. Some Transfinite Ordinal Numbers
 2. Comparing the Sizes of Sets
 3. Ordinal and Cardinal Numbers
 4. Cantor's Diagonal Argument
 5. Cantor's Power Set Argument
 5.1 Sketch of the Power Set Argument
 5.2 The Power Set Argument in Detail
 5.3 The Beth Numbers
 6. The Aleph Numbers
 7. Transfinite Recursion
 7.1 Rules for the Long Line
 7.2 The Sequence of Universes
 7.3 Degrees of Divine Perfection
Further Study
Glossary of Symbols
References
Index
