# Book:Gaisi Takeuti/Introduction to Axiomatic Set Theory

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

Published $1971$, Springer-Verlag

ISBN 0 387 05302 6.

### 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. 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
4.15. 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 Specification
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 Cartesian Product with Proper Class is Proper Class
6.10. Definition:Unique
6.11. Definition:Image (Set Theory)/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
7.14. 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

8.4. 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
8.13. 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 Class
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 of Finite Set
10.50. Set of All Mappings of Cartesian Product
10.51. Definition:Cofinal Relation on Ordinals
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

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

$\S 13$. 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