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. 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
 * 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

 * 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