Book:Gaisi Takeuti/Introduction to Axiomatic Set Theory

Subject Matter

 * Set Theory
 * Inner Model Theory

Contents

 * Preface

$\S 2$. Language and Logic

 * 2. Definition:Language of Set Theory

$\S 3$. Equality

 * 3.1. Definition:Set Equality
 * 3.2. Equality is Reflexive and Equality is Symmetric and Equality is Transitive
 * 3.3. Substitution of Elements
 * 3.4. Substitutivity of Equality

$\S 4$. Classes

 * 4.1–4.4. Definition:Class
 * 4.5. Definition:Class Equality
 * 4.6. Definition:Class Membership
 * 4.7. Class Equality is Reflexive and Class Equality is Symmetric and Class Equality is Transitive
 * 4.8. Substitutivity of Class Equality
 * 4.9. Class Equal to All its Elements
 * 4.10. Definition:Small Class and Definition:Proper Class
 * 4.11. Set is Small Class
 * 4.12. Class Member of Class Builder
 * 4.13. Definition:Russell Class
 * 4.14. Russell's Paradox
 * 4.15. Definition:Definable

$\S 5$. The Elementary Properties of Classes

 * 5.1. Definition:Singleton and Definition:Doubleton
 * 5.2. Definition:Ordered Pair
 * 5.3. Definition:Unordered Tuple
 * 5.4. Definition:Ordered Tuple
 * 5.5. Definition:Set Union/General Definition
 * 5.6. Definition:Union and Definition:Intersection
 * 5.7. Union of Doubleton
 * 5.8. Union is a Set
 * 5.9. Definition:Subset and Definition:Proper Subset
 * 5.10. Definition:Power Set
 * 5.11. Axiom:Axiom of Subsets
 * 5.12. Axiom of Subsets Equivalents
 * 5.13. Axiom of Subsets Equivalents
 * 5.14. Definition:Set Difference
 * 5.15. Set Difference is Set
 * 5.16. Definition:Empty Set
 * 5.17. Set Difference with Self is Empty Set
 * 5.18. Empty Set Exists
 * 5.19. Nonempty Class has Members
 * 5.20. No Membership Loops
 * 5.21. Class is Not Element of Itself
 * 5.22. Definition:Universal Class
 * 5.23. Universal Class is Proper
 * 5.24. Epsilon Induction

$\S 6$. Functions and Relations

 * 6.1. Definition:Cartesian Product
 * 6.2. Cartesian Product is Small
 * 6.3. Definition:Inverse Relation
 * 6.4. Definition:Relation and Definition:Injective and Definition:Mapping
 * 6.5. Definition:Domain (Set Theory) and Definition:Range
 * 6.6. Definition:Restriction and Definition:Image and Definition:Composition of Relations
 * 6.7. Image is Small
 * 6.8. Inverse is Small and Domain is Small and Range is Small
 * 6.9. Cartesian Product is Small iff Inverse is Small and Proper Class Means Proper Cartesian Product
 * 6.10. Definition:Unique
 * 6.11. Definition:Image/Relation/Element/Singleton
 * 6.12. Uniqueness Condition for Relation Value
 * $6.13$. Value of Relation is Small
 * $6.14$. Definition:Mapping
 * $6.15$. Mapping is Small
 * $6.16$. Restriction is Small
 * $6.17$. Definition:Relation
 * $6.18$. Definition:Poset and Definition:Strict Total Ordering
 * $6.19$. Preimage of Singleton
 * $6.20$. Definition:Preimage
 * $6.21$. Definition:Foundational Relation
 * $6.22$. Definition:Epsilon Relation
 * $6.23$. Foundational Relation has no Relational Loops
 * $6.24$. Definition:Strict Well-Ordering
 * $6.25$. Well-Ordering is Total Ordering
 * $6.26$. Well-Founded Relation Determines Minimal Elements/Special Case
 * $6.27$. Well-Founded Induction/Special Case
 * $6.28$. Definition:Order Isomorphism
 * $6.29$. Definition:Identity Mapping
 * $6.30$. Identity Mapping is Order Isomorphism and Inverse of Order Isomorphism and Composite of Order Isomorphisms
 * $6.31$. Order Isomorphism Preserves Minimal Elements and Order Isomorphism Preserves Initial Segments
 * $6.32$. Order Isomorphism on Foundational Relations and Order Isomorphism on Well-Ordered Sets
 * $6.33$. Induced Relation Generates Order Isomorphism

$\S 7$. 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 is 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/General Definition
 * $7.38$. 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 Unique Ordinal under Unique Mapping
 * $7.52$. 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

 * $8.1$. Definition:Ordinal Addition
 * $8.2$. Ordinal Addition is Closed
 * $8.3$. Ordinal Addition by Zero
 * $8.4$. Membership Left Compatible with Ordinal Addition
 * $8.5$. Ordinal Addition is Left Cancellable
 * $8.6$. Supremum Inequality for Ordinals
 * $8.7$. Subset Right Compatible with Ordinal Addition
 * $8.8$. Ordinal Subtraction when Possible is Unique
 * $8.9$. Natural Number Addition is Closed
 * $8.10$. Natural Number Plus Ordinal
 * $8.11$. Limit Ordinals Preserved Under Ordinal Addition
 * $8.12$. Ordinal Addition is Associative
 * $8.13$. Unique Limit Ordinal Plus Unique Natural Number
 * $8.14$. Definition:Ordinal Subtraction
 * $8.15$. Definition:Ordinal Multiplication
 * $8.16$. Ordinal Multiplication is Closed
 * $8.17$. Natural Number Multiplication is Closed
 * $8.18$. Ordinal Multiplication by Zero and Ordinal Multiplication by One
 * $8.19$. Membership Left Compatible with Ordinal Multiplication
 * $8.20$. Ordinal Multiplication is Left Cancellable
 * $8.21$. Subset Right Compatible with Ordinal Multiplication
 * $8.22$. Ordinals have No Zero Divisors
 * $8.23$. Limit Ordinals Preserved Under Ordinal Multiplication
 * $8.24$. Ordinal Less than Ordinal Times Limit
 * $8.25$. Ordinal Multiplication is Left Distributive
 * $8.26$. Ordinal Multiplication is Associative
 * $8.27$. Division Theorem for Ordinals
 * $8.28$. Division Theorem
 * $8.29$. Natural Number Times Ordinal
 * $8.30$. Definition:Ordinal Exponentiation
 * $8.31$. Exponent Base of One
 * $8.32$. Exponent Not Equal to Zero
 * $8.33$. Membership Left Compatible with Ordinal Exponentiation
 * $8.34$. Membership Left Compatible with Ordinal Exponentiation
 * $8.35$. Subset Right Compatible with Ordinal Exponentiation
 * $8.36$. Condition for Membership Right Compatible with Ordinal Exponentiation
 * $8.37$. Lower Bound for Ordinal Exponentiation
 * $8.38$. Unique Ordinal Exponentiation Inequality
 * $8.39$. Limit Ordinals Closed under Ordinal Exponentiation
 * $8.40$. Ordinal Less Than Ordinal to Limit Power
 * $8.41$. Ordinal Sum of Powers
 * $8.42$. Ordinal Power of Power
 * $8.43$. Upper Bound of Ordinal Sum
 * $8.44$. Basis Representation Theorem for Ordinals and Definition:Cantor Normal Form
 * $8.45$. Ordinal Multiplication via Cantor Normal Form/Infinite Exponent
 * $8.46$. Ordinal Multiplication via Cantor Normal Form/Limit Base
 * $8.47$. Ordinal Exponentiation of Terms
 * $8.48$. Inequality for Ordinal Exponentiation
 * $8.49$. Ordinal Exponentiation via Cantor Normal Form/Limit Exponents
 * $8.50$. Ordinal Exponentiation via Cantor Normal Form/Corollary

$\S 9$. Relational Closure and the Rank Functions

 * $9.1$. Transitive Closure Always Exists (Set Theory)
 * $9.2$. Definition:Transitive Closure (Set Theory)
 * $9.3$. Relational Closure Exists for Well-Founded Sets
 * $9.4$. Well-Founded Relation Determines Minimal Elements/Special Case 2
 * $9.5$. Definition:Closed Relation and Definition:Closure (Abstract Algebra)/Algebraic Structure
 * $9.6$. Closure for Any Finite List of Relations and Operations
 * $9.7$. Well-Founded Recursion
 * $9.8$. Definition:Supertransitive
 * $9.9$. Definition:Von Neumann Hierarchy
 * $9.10$. Von Neumann Hierarchy is Supertransitive and Von Neumann Hierarchy Comparison
 * $9.11$. Definition:Well-Founded Set
 * $9.13$. Every Set in Von Neumann Universe
 * $9.14$. Definition:Rank (Set Theory)
 * $9.15$. Rank is Ordinal and Ordinal Equal to Rank and Ordinal Subset of Rank
 * $9.16$. Membership Rank Inequality
 * $9.17$. Rank of Set Determined by Members
 * $9.18$. Rank of Ordinal
 * $9.19$. Bounded Rank implies Small Class
 * $9.20$. Axiom of Foundation (Strong Form)
 * $9.21$. Well-Founded Relation Determines Minimal Elements
 * $9.22$. Well-Founded Induction

$\S 10$. Cardinal Numbers

 * $10.1$. Definition:Set Equivalence
 * $10.2$. Set Equivalence is Equivalence Relation
 * $10.3$. Cantor-Bernstein-Schroeder Theorem
 * $10.4$. Cantor's Theorem
 * $10.5$. Cantor's Theorem
 * $10.6$. Power Sets are Equinumerous
 * $10.7$. Definition:Cardinal Number
 * $10.8$. Cardinal Number is Ordinal
 * $10.9$. Condition for Set Equivalent to Associated Cardinal Number
 * $10.10$. Cardinal Number Equivalent to Empty Set
 * $10.11$. Ordinal Number Equivalent to Cardinal Number
 * $10.12$. Cardinal Number Less than Ordinal
 * $10.13$. Cardinal Number Less than Ordinal/Corollary
 * $10.14$. Equivalent Sets have Equal Cardinal Numbers
 * $10.15$. Condition for Set Union Equivalent to Associated Cardinal Number and Condition for Cartesian Product Equivalent to Associated Cardinal Number
 * $10.16$. Cardinal of Cardinal Equal to Cardinal
 * $10.17$. Equality of Natural Numbers
 * $10.18$. Pigeonhole Principle
 * $10.19$. Cardinal of Natural Number
 * $10.20$. Ordinal Equivalent to Natural Number
 * $10.21$. Definition:Finite Set and Definition:Infinite Set
 * $10.22$. Subset implies Cardinal Inequality
 * $10.23$. Subset of Ordinal implies Cardinal Inequality
 * $10.24$. Subset of Finite Set is Finite
 * $10.25$. Set Less than Cardinal Product
 * $10.26$. Image Cardinal Inequality
 * $10.27$. Surjection iff Cardinal Inequality
 * $10.28$. Cardinal of Union Less than Cardinal of Cartesian Product
 * $10.29$. Union of Finite Sets is Finite and Product of Finite Sets is Finite
 * $10.30$. Ordinal is Finite iff Natural Number
 * $10.31$. Cardinal Inequality implies Ordinal Inequality
 * $10.32$. Cardinal Number Plus One Less than Cardinal Product
 * $10.33$. Cardinal Product Equal to Cardinal
 * $10.34$. Cardinal Product Equal to Cardinal/Corollary
 * $10.35$. Cardinal Product Equal to Maximum and Cardinal of Union Equal to Maximum
 * $10.36$. Definition:Class of Cardinals
 * $10.37$. Cardinal Class Subset of Ordinal Class
 * $10.38$. Cardinal of Cardinal Equal to Cardinal/Corollary
 * $10.39$. Class of Cardinals Contains Minimal Infinite Successor Set
 * $10.40$. Cardinal Equal to Collection of All Dominated Ordinals
 * $10.41$. Cardinal Class is Proper Class
 * $10.42$. Definition:Infinite Cardinal Class
 * $10.43$. Infinite Cardinal Class is Proper Class
 * $10.44$. Definition:Aleph Function
 * $10.45$. Definition:Aleph Function
 * $10.46$. Ordinal in Aleph iff Cardinal in Aleph and Aleph Product is Aleph and Surjection from Aleph to Ordinal
 * $10.47$. Definition:Set of All Mappings
 * $10.48$. Cardinal Exponentiation is a Small Class
 * $10.49$. Cardinality of Power Set
 * $10.50$. Set of All Mappings of Cartesian Product
 * $10.51$. Definition:Cofinal Ordinal
 * $10.52$. Cofinal Ordinal Relation is Reflexive and Cofinal Ordinal Relation is Transitive
 * $10.53$. Cofinal to Zero iff Ordinal is Zero and Condition for Cofinal Nonlimit Ordinals
 * $10.54$. Nonlimit Ordinal Cofinal to One
 * $10.55$. Cofinal Limit Ordinals
 * $10.56$. Subset of Ordinal is Cofinal
 * $10.57$. Subset of Ordinal is Cofinal/Corollary
 * $10.58$. Condition for Cofinal Limit Ordinals
 * $10.59$. Limit Ordinal Cofinal with its Aleph
 * $10.60$. Ordinal Cofinal to Two Ordinals implies Cofinal to Subset of Ordinal
 * $10.61$. Definition:Cofinality
 * $10.62$. Cofinality is Cardinal
 * $10.63$. Cofinality of Infinite Cardinal is Infinite Cardinal
 * $10.64$. Cofinality of Ordinal is Cofinality of Aleph
 * $10.65$. Definition:Regular Cardinal and Definition:Singular Cardinal
 * $10.66$. Definition:Weakly Inaccessible and Definition:Strongly Inaccessible
 * $10.67$. Weakly Inaccessible Cardinals are Aleph Fixed Points
 * $10.68$. Union of Cardinals is Cardinal
 * $10.69$. Union of Infinite Cardinals is Infinite Cardinal
 * $10.70$. Aleph Fixed Point Exists

$\S 11$. The Axiom of Choice, the Greater Continuum Hypothesis, and Cardinal Arithmetic

 * $11.1$. Definition:Chain (Set Theory) and Definition:Maximal Element
 * $11.2$. Well-Ordering Theorem and Zorn's Lemma and Cantor's Law of Trichotomy
 * $11.3$. Set Equivalent to Some Ordinal
 * $11.4$. Set Equivalent to Cardinal
 * $11.5$. Subset implies Cardinal Inequality
 * $11.6$. Set Less than Cardinal Product
 * $11.7$. Cardinal of Image Less than Cardinal
 * $11.8$. Cantor-Bernstein-Schroeder Theorem
 * $11.9$. Cardinal Less than Cardinal of Powerset

$\S 12$. Models

 * $12.1$. Definition:Structure (Set Theory)
 * $12.2$. Definition:Standard Structure
 * $12.3$. Definition:Standard Structure
 * $12.4$. Definition:Relativisation
 * $12.5$. Relativisation is Standard Model
 * $12.6$. Model Satisfies Axioms implies Model Satisfies Theorems
 * $12.7$. Definition:Standard Transitive Model

Additional sections

 * $\S 13$. 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