# Book:Gaisi Takeuti/Introduction to Axiomatic Set Theory

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## Contents

- 1 Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*- 1.1 Subject Matter
- 1.2 Contents
- 1.2.1 $\S 1$. Introduction
- 1.2.2 $\S 2$. Language and Logic
- 1.2.3 $\S 3$. Equality
- 1.2.4 $\S 4$. Classes
- 1.2.5 $\S 5$. The Elementary Properties of Classes
- 1.2.6 $\S 6$. Functions and Relations
- 1.2.7 $\S 7$. Ordinal Numbers
- 1.2.8 $\S 8$. Ordinal Arithmetic
- 1.2.9 $\S 9$. Relational Closure and the Rank Functions
- 1.2.10 $\S 10$. Cardinal Numbers
- 1.2.11 $\S 11$. The Axiom of Choice, the Greater Continuum Hypothesis, and Cardinal Arithmetic
- 1.2.12 $\S 12$. Models
- 1.2.13 Additional sections

## Gaisi Takeuti and Wilson M. Zaring: *Introduction to Axiomatic Set Theory*

Published $1971$, **Springer-Verlag**

- ISBN 0 387 05302 6.

### Subject Matter

### Contents

- Preface

#### $\S 1$. Introduction

#### $\S 2$. Language and Logic

#### $\S 3$. Equality

#### $\S 4$. Classes

- 4.1–4.4. 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:Union of Set of Sets
- 5.6. Definition:Set Union and Definition:Set Intersection
- 5.7. Union of Doubleton
- 5.8. Union of Small Classes is Small
- 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 (Set Theory) and Definition:Composition of Relations
- 6.7. Image of Small Class under Mapping is Small
- 6.8. Inverse of Small Relation is Small and Domain of Small Relation is Small and Range of Small Relation 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 whose Domain is Small Class is Small
- 6.16. Restriction of Mapping to Small Class is Small
- 6.17. Definition:Relation
- 6.18. Definition:Partially Ordered Set 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. Proper Well-Ordering Determines Smallest Elements
- 6.27. Well-Ordered Induction
- 6.28. Definition:Order Isomorphism
- 6.29. Definition:Identity Mapping
- 6.30. Identity Mapping is Order Isomorphism and Inverse of Order Isomorphism is Order Isomorphism and Composite of Order Isomorphisms is Order Isomorphism
- 6.31. Order Isomorphism Preserves Minimal Elements and Order Isomorphism Preserves Initial Segments
- 6.32. Order Isomorphism on Foundational Relation preserves Foundational Structure and Order Isomorphism on Well-Ordered Set preserves Well-Ordering
- 6.33. Induced Relation Generates Order Isomorphism

#### $\S 7$. Ordinal Numbers

- 7.1. Definition:Transitive Class
- 7.2. Element of Transitive Class
- 7.3. Equivalence of Definitions of Ordinal
- 7.4. Equivalence of Definitions of Ordinal
- 7.5. Subset of Ordinals has Minimal Element
- 7.6. Initial Segment of Ordinal is Ordinal
- 7.7. Transitive Set is Proper Subset of Ordinal iff Element of Ordinal
- 7.8. Transitive Set is Proper Subset of Ordinal iff Element of Ordinal/Corollary
- 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 is Member of Ordinal Class
- 7.15. Ordinal is 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 is 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 forms Peano Structure
- 7.31. Principle of Mathematical Induction for 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:Intersection of Set of Sets
- 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/Proof 2
- 7.44. Definition:Ordinal Function
- 7.45. Well-Ordered Transitive Subset is 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
- 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 Initial Segment of Canonical Order is Set
- 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 is Left Compatible with Ordinal Addition
- 8.5. Ordinal Addition is Left Cancellable
- 8.6. Supremum Inequality for Ordinals
- 8.7. Subset is Right Compatible with Ordinal Addition
- 8.8. Ordinal Subtraction when Possible is Unique
- 8.9. Natural Number Addition is Closed
- 8.10. Finite Ordinal Plus Transfinite Ordinal
- 8.11. Limit Ordinals Preserved Under Ordinal Addition
- 8.12. Ordinal Addition is Associative
- 8.13. Unique Limit Ordinal Plus Unique Finite Ordinal
- 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 is Left Compatible with Ordinal Multiplication
- 8.20. Ordinal Multiplication is Left Cancellable
- 8.21. Subset is Right Compatible with Ordinal Multiplication
- 8.22. Ordinals have No Zero Divisors
- 8.23. Limit Ordinals Preserved Under Ordinal Multiplication
- 8.24. Ordinal is 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. Finite Ordinal Times Ordinal
- 8.30. Definition:Ordinal Exponentiation
- 8.31. Exponent Base of One
- 8.32. Exponent Not Equal to Zero
- 8.33. Membership is Left Compatible with Ordinal Exponentiation
- 8.34. Membership is Left Compatible with Ordinal Exponentiation
- 8.35. Subset is Right Compatible with Ordinal Exponentiation
- 8.36. Condition for Membership is 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 is 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 Set-Like Relation
- 9.4. Well-Founded Proper Relational Structure Determines Minimal Elements
- 9.5. Definition:Closed Relation and Definition:Closure (Abstract Algebra)/Algebraic Structure
- 9.6. Closure for Finite Collection 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 is Subset of Rank of Small Class iff Not in Von Neumann Hierarchy
- 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-Schröder 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 Cardinal Number
- 10.10. Cardinal Number Equivalence or Equal to Universe
- 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 Finite Ordinal
- 10.20. Finite Ordinal is equal 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. Cardinality of Image of Mapping not greater than Cardinality of Domain
- 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/Proof 2 and Product of Finite Sets is Finite/Proof 2
- 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. Non-Finite Cardinal is equal to Cardinal Product
- 10.34. Non-Finite Cardinal is equal to Cardinal Product/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 is 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 Mapping
- 10.45. Definition:Aleph Mapping#Notation
- 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. Set of All Mappings is 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 Cardinal and Definition:Strongly Inaccessible Cardinal
- 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-Schröder Theorem
- 11.9. Cardinal Less than Cardinal of Powerset

#### $\S 12$. Models

#### 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