Book:Gaisi Takeuti/Introduction to Axiomatic Set Theory

Subject Matter

 * Set Theory
 * Inner Model Theory

Contents

 * Preface


 * $\S 1$. Introduction
 * $\S 2$. Language and Logic
 * $\S 3$. Equality
 * $\S 4$. Classes
 * $\S 5$. The Elementary Properties of Classes
 * $5.20$. No Membership Loops
 * $\S 6$. Functions and Relations
 * $6.23$. Foundational Relation has no Relational Loops
 * $6.26$. Well-Founded Relation Determines Minimal Elements/Special Case
 * $6.27$. Well-Founded Induction/Special Case
 * $\S 7$. Ordinal Numbers
 * $7.1$. Definition:Transitive Class
 * $7.2$. Element of Transitive Class
 * $7.3$. Alternate Definition of an Ordinal
 * $7.4$. Alternate Definition of an Ordinal
 * $7.5$. Subset of Ordinals has Minimal Element
 * $7.6$. Initial Segment of Ordinal is Ordinal
 * $7.7$. Ordinal Proper Subset Membership
 * $7.8$. Ordinal Proper Subset Membership
 * $7.9$. Intersection of Two Ordinals is Ordinal
 * $7.10$. Ordinal Membership Trichotomy
 * $7.11$. Definition:Ordinal Class
 * $7.12$. Ordinal Class is Ordinal
 * $7.13$. Burali-Forti Paradox
 * $7.14$. Ordinal Member of Ordinal Class
 * $7.15$. Ordinal Subset of Ordinal Class
 * $7.17$. Transfinite Induction
 * $7.19$. Union of Subset of Ordinals is Ordinal
 * $7.20$. Union of Ordinals is Least Upper Bound
 * $7.21$. Union of Ordinals is Least Upper Bound
 * $7.22$. Definition:Successor Set
 * $7.23$. Ordinal Less than Successor
 * $7.24$. Successor Set of Ordinal is Ordinal
 * $7.25$. No Ordinal Between Set and Successor
 * $7.26$. No Largest Ordinal
 * $7.27$. Definition:Limit Ordinal
 * $7.28$. Definition:Minimal Infinite Successor Set
 * $7.30$. Minimal Infinite Successor Set Fulfils Peano Axioms
 * $7.31$. Principle of Finite Induction/Minimal Infinite Successor Set‎
 * $7.32$. Minimal Infinite Successor Set is Ordinal
 * $7.33$. Minimal Infinite Successor Set is Limit Ordinal
 * $7.34$. No Infinitely Descending Membership Chains
 * $7.35$. Definition:Set Intersection
 * $7.38$. Isomorphic Ordinals are Equal
 * $7.39$. Ordinals Isomorphic to the Same Well-Ordered Set
 * $7.40$. Transfinite Recursion/Theorem 1
 * $7.41$. Transfinite Recursion/Corollary
 * $7.42$. Transfinite Recursion/Theorem 2
 * $7.43$. Principle of Recursive Definition/Minimal Infinite Successor Set
 * $7.44$. Definition:Ordinal Function
 * $7.45$. Well-Ordered Transitive Subset Equal or Equal to Initial Segment
 * $7.46$. Condition for Injective Mapping on Ordinals
 * $7.47$. Maximal Injective Mapping from Ordinals to a Set
 * $7.48$. Order Isomorphism between Ordinals and Proper Class/Lemma
 * $7.49$. Order Isomorphism between Ordinals and Proper Class/Theorem
 * $7.50$. Order Isomorphism between Ordinals and Proper Class/Corollary
 * $7.51$. Strict Well-Ordering Isomorphic to a Unique Ordinal Under Unique Function
 * $7.52$. Unique Isomorphism between Ordinal Subset and Unique Ordinal
 * $7.53$. Definition:Lexicographic Order
 * $7.54$. Lexicographic Order Forms Well-Ordering on Ordered Pairs of Ordinals and Lexicographic Order Initial Segments
 * $7.55$. Definition:Canonical Order
 * $7.56$. Canonical Order Well-Orders Ordered Pairs of Ordinals and Canonical Order Initial Segments
 * $7.57$. Definition:Canonical Order
 * $\S 8$. Ordinal Arithmetic
 * $\S 9$. Relational Closure and the Rank Functions
 * $\S 10$. Cardinal Numbers
 * $\S 11$. The Axiom of Choice, the Greater Continuum Hypothesis, and Cardinal Arithmetic
 * $\S 12$. Models
 * $\S 13$. Absoluteness
 * $\S 14$. The Fundamental Operations
 * $\S 15$. The Gödel Model
 * $\S 16$. The Arithmetization of Model Theory
 * $\S 17$. Cohen's Model
 * $\S 18$. Forcing
 * $\S 19$. Languages, Structures and Models


 * Bibliography


 * Problem List


 * Appendix


 * Index


 * Index of Symbols