Book:Gaisi Takeuti/Introduction to Axiomatic Set Theory

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

Published $\text {1971}$, Springer-Verlag

ISBN 0 387 05302 6

Subject Matter

• Set Theory
• Inner Model Theory

Contents

This page has been identified as a candidate for refactoring of medium complexity. In particular: Complete rewrite needed so as to be compatible with book contents as used elsewhere in $\mathsf{Pr} \infty \mathsf{fWiki}$. Until this has been finished, please leave {{Refactor}} in the code. New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Refactor}} from the code.

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. Characterization of 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 is Extensional

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 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 of Relation and Definition:Range of Relation

6.6. Definition:Restriction and Definition:Image (Relation 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 (Relation 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:Strictly Well-Founded Relation

6.22. Definition:Epsilon Relation

6.23. Strictly Well-Founded 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 Strictly Minimal Elements and Order Isomorphism Preserves Initial Segments

6.32. Order Isomorphism on Strictly Well-Founded Relation preserves Strictly Well-Founded 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:Class of All Ordinals

7.12. Class of All Ordinals is Ordinal

7.13. Burali-Forti Paradox

7.14. Ordinal is Member of Class of All Ordinals

7.15. Ordinal is Subset of Class of All Ordinals

7.17. Transfinite Induction

7.19. Union of Set 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 Natural Number between Number and Successor

7.26. No Largest Ordinal

7.27. Definition:Limit Ordinal

7.28. Definition:Minimally Inductive Set

7.30. Minimally Inductive Set forms Peano Structure

7.31. Principle of Mathematical Induction for Minimally Inductive Set

7.32. Minimally Inductive Set is Ordinal

7.33. Minimally Inductive 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. First Principle of Transfinite Recursion

7.41. Transfinite Recursion Theorem/Corollary

7.42. Second Principle of Transfinite Recursion

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 Initial Segment of Ordinals under Lexicographic Order

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 of Set Always Exists

9.2. Definition:Transitive Closure of Set

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. Set has Rank

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. Strictly Well-Founded Relation determines Strictly Minimal Elements

9.22. Well-Founded Induction

$\S 10$. Cardinal Numbers

10.1. Definition:Set Equivalence

10.2. Set Equivalence behaves like Equivalence Relation

10.3. Cantor-Bernstein-Schröder Theorem

10.4. Cantor's Theorem

10.5. Cantor's Theorem

10.6. Power Sets of Equinumerous 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 All Cardinals

10.37. Class of All Cardinals is Subclass of Class of All Ordinals

10.38. Cardinal of Cardinal Equal to Cardinal/Corollary

10.39. Class of All Cardinals Contains Minimally Inductive Set

10.40. Cardinal Equal to Collection of All Dominated Ordinals

10.41. Class of All Cardinals is Proper Class

10.42. Definition:Class of Infinite Cardinals

10.43. Class of Infinite Cardinals 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 (Order Theory) and Definition:Maximal Element

11.2. Zermelo's 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