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## Cardinal and ordinal numbers / Waclaw Sierpinski.

Idioma: Inglés Lenguaje original: Polaco Series Monografie matematyczne: t. 34.Editor: Warszawa : Panstwowe Wydawn. Naukowe, 1958Descripción: 487 p. ; 25 cmOtra clasificación: 03E10
Contenidos:
```Chapter I. Sets and elementary set operations
1. Sets 
2. Elements of a set 
3. Symbols E and not E 
4. Set consisting of one element 
5. The empty set 
6. Equality of sets 
7. Sets of sets 
8. Subset of a set 
9. Sum of sets 
10. Difference of sets 
11. Product of sets 
12. Disjoint sums 
13. Complement of a set 
14. Ordered pairs 
, 15. Correspondence. Function 
16. One-to-one correspondence 
17. Cartesian product of sets 
18. Exponentiation of sets 
Chapter II. Equivalent sets
1. Equivalent sets. Relation ~ 
2. Finite and infinite sets 
3. Fundamental properties of the relation ~ 
4. Effectively equivalent sets 
5. Various theorems on the equivalence of sets 
6. The Cantor-Bernstein Theorem 
Chapter III. Denumerable and non-denumerable sets
1. Denumerable and effectively denumerable sets 
2. Effective denumerability of the set of all rational numbers 
3. Effective denumerability of the infinite set of non-overlapping intervals 
4. Effective denumerability of the set of all finite sequences of rational numbers 
5. Effective denumerability of the set of all algebraic numbers 
6. Non-denumerable sets 
1. Properties of sets containing denumerable subsets 
8. Sets infinite in the sense of Dedekind 
9. Various definitions of finite sets 
10. Denumerability of the Cartesian product of two denumerable sets 
Chapter TV. Sets of the power of the continuum
1. Sets of the power of the oontinuum and sets effectively of the power of the continuum 
2. Non-denumerability of the set of real numbers 
3. Removing a denumerable set from a set of the power of the continuum 
4. Sot of real numbers of an arbitrary interval 
5. Sum of two sets of the power of the continuum 
6. Cartesian product of a denumerable set and a set of the power of the continuum 
7. Set of all infinite sequences of natural numbers 
8. Cartesian product of two sets of the power of the continuum 
9. Impossibility of a continuous (1-1) mapping of a plane on a straight line 
10. Continuous curve filling up a square 
11. Set of all infinite sequences of real numbers 
12. Continuous curve filling up a denumerably dimensional cube 
13. Set of all continuous functions 
14. Decomposition of a set of natural numbers into a continuum of almost disjoint sets 
Chapter V. Comparing the power of sets
1. Sets of less and greater power 
2. Sets of greater power than finite sets and denumerable sets. Hypothesis of the continuum 
3. Cantor’s theorem on the set of all subsets of a given set 
4. Generalized Continuum Hypothesis 
5. Forming sets of ever greater powers 
Chapter VI. Axiom of choice
1. The axiom of choice. Controversy about it 
2. The axiom of choice for a finite set of sets 
2a. The axiom of choice for an infinite sequence of sets 
3. Hilbert’s axiom 
4. General principle of choice 
5. Axiom of choice for finite sets 
6. Examples of cases where we are able or not able to make an effective choice 
7. Applications of the axiom of choice 
8. The m-to-n correspondence 
9. Dependent choices 
Chapter VII. Cardinal numbers and operations on them
1. Cardinal numbers 
2. Sum of cardinal numbers 
3. Product of two cardinal numbers 
4. Exponentiation of cardinal numbers 
5. Power of the set of all subsets of a given set 
Chapter VIII. Inequalities for cardinal numbers
1. Definition of an inequality between two cardinal numbers 
2. Transitivity of the relation of inequality. Addition of inequalities 
3. Exponentiation of inequalities for cardinal numbers 
4. Relation m ≤ * n 
Chapter IX. Difference of cardinal numbers
1. Theorem of A. Tarski and F. Bernstein 
2. Theorem on increasing the diminuend 
3. Theorem on increasing the subtrahend 
4. Difference in which the subtrahend is a natural number 
5. Proof of formula 2m —m = 2m for m ≥ ϰ0 without the aid of the axiom of choice 
6. Quotient of cardinal numbers 
Chapter X. Infinite series and infinite products of cardinal numbers
1. Sum of an infinite series of cardinal numbers 
2. Properties of an infinite series of cardinal numbers 
3. Examples of infinite series of cardinal numbers 
4. Sum of an arbitrary set of cardinal numbers 
5. Infinite product of cardinal numbers 
6. Properties of infinite products of cardinal numbers. Examples 
7. Theorem of J. König 
8. Product of an arbitrary set of cardinal numbers 
Chapter XI. Ordered sets
1. Ordered sets 
2. Partially ordered sets 
3. Lattices 
4. Similarity of sets 
5. First and last element of an ordered set. Cuts. Jumps. Density and continuity of a set 
6. Finite ordered sets 
7. Sets of type ω 
8. Sets of type ŋ 
9. Dense ordered sets as subsets of continuous sets 
10. Sets of type λ 
Chapter XII. Order types and operations on them
1. Order types 
2. Sum of two ordered types 
3. Product of two order types 
4. Sum of an infinite series of order types 
5. Power of the set of all denumerable order types 
6. Power of the set of all order types of the power of the continuum 
7. Sum of an arbitrary ordered set of order types 
8. Infinite products of order types 
9. Segments and remainders of order types 
10. Divisors of order types 
11. Comparison of order types 
Chapter XIII. Well-ordered sets
1. Well-ordered sets 
2. The principle of transfinite induction 
3. Induction for ordered sets 
4. Similar mapping of a well-ordered set on its subset 
5. Properties of segments of well-ordered sets 
Chapter XIV. Ordinal numbers
1. Ordinal numbers. Ordinal numbers as indices of the elements of well-ordered sets 
2. Sets of ordinal numbers 
3. Sum of ordinal numbers 
4. Properties of the sum of ordinal numbers. Numbers of the. 1-st and of the 2-nd kind 
5. Remainders of ordinal numbers 
6. Prime components 
7. Transfinite sequences of ordinal numbers and their limits 
8. Infinite series of ordinal numbers and their sums 
9. Product of ordinal numbers 
10. Properties of the product of ordinal numbers 
11. Theorem on the division of ordinal numbers 
12. Divisors of ordinal numbers 
13. Prime factors of ordinal numbers 
14. Certain properties of prime components 
15. Exponentiation of ordinal numbers 
16. Definitions by transfinite induction 
17. Transfinite products of ordinal numbers 
18. Properties of the powers of ordinal numbers 
19. The power ωa. Normal expansions of ordinal numbers 
20. Epsilon numbers 
21. Applications of the normal form 
22. Determination of all cardinal numbers that are prime factors 
23. Expanding ordinal numbers into prime factors 
24. Roots of ordinal numbers 
25. On ordinal numbers commutative with respect to addition 
26. On ordinal numbers commutative with respect to multiplication 
27. On the equation aB = Ba for ordinal numbers 
28. Natural sum and natural product of ordinal numbers 
29. Exponentiation of order types 
Chapter XV. Number classes and alephs
1. Numbers of the 1-st and of the 2-nd class 
2. Cardinal number ϰ1 
3. ϰ1— 2X° hypothesis 
4. Properties of ordinal numbers of the 2-nd class 
5. Transfinite induction for numbers of the 1-st class and of the 2-nd class 
6. Convergence and limit of transfinite sequences of real numbers 
7. Initial numbers, alephs and their notation 
8. Formula ϰ2a = ϰa and conclusions from it 
9. A proposition of elementary geometry, equivalent to the continuum hypothesis 
10. Difference of alephs. Sums and products of transfinite sequences of successive alephs 
11. Limit of transfinite sequences of initial numbers. Regular and singular initial numbers. Inaccessible alephs 
Chapter XVI. Zermelo’s theorem and other theorems equivalent to the axiom of choice
1. Equivalence of the axiom od choice, of Zermelo’s theorem and of the problem of trichotomy 
2. Various theorems on cardinal numbers equivalent to the axiom of choice 
3. A. Lindenbaum’s theorem 
4. Equivalence of the axiom of choice to the theorems of Zorn and Teichmüller 
5. Inference of the axiom of choice from the generalized continuum hypothesis 
Chapter XVII. Applications of Zermelo’s theorem
1. Hamel’s basis 
2. Plane set having exactly two points in common with every straight line 
3. Decomposition of an infinite set of power tn into more than tn almost disjoint sets 
4. Some theorems on families of subsets of sets of given power 
5. The power of the set of all order types of a given power 
6. Applications of Zermelo’s theorem to the theory of ordered sets 
Appendix 
Bibliography 
Index ``` Average rating: 0.0 (0 votes)
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Bibliografía: p. -481.

Chapter I. Sets and elementary set operations --
1. Sets  --
2. Elements of a set  --
3. Symbols E and not E  --
4. Set consisting of one element  --
5. The empty set  --
6. Equality of sets  --
7. Sets of sets  --
8. Subset of a set  --
9. Sum of sets  --
10. Difference of sets  --
11. Product of sets  --
12. Disjoint sums  --
13. Complement of a set  --
14. Ordered pairs  --
, 15. Correspondence. Function  --
16. One-to-one correspondence  --
17. Cartesian product of sets  --
18. Exponentiation of sets  --
Chapter II. Equivalent sets --
1. Equivalent sets. Relation ~  --
2. Finite and infinite sets  --
3. Fundamental properties of the relation ~  --
4. Effectively equivalent sets  --
5. Various theorems on the equivalence of sets  --
6. The Cantor-Bernstein Theorem  --
Chapter III. Denumerable and non-denumerable sets --
1. Denumerable and effectively denumerable sets  --
2. Effective denumerability of the set of all rational numbers  --
3. Effective denumerability of the infinite set of non-overlapping intervals  --
4. Effective denumerability of the set of all finite sequences of rational numbers  --
5. Effective denumerability of the set of all algebraic numbers  --
6. Non-denumerable sets  --
1. Properties of sets containing denumerable subsets  --
8. Sets infinite in the sense of Dedekind  --
9. Various definitions of finite sets  --
10. Denumerability of the Cartesian product of two denumerable sets  --
Chapter TV. Sets of the power of the continuum --
1. Sets of the power of the oontinuum and sets effectively of the power of the continuum  --
2. Non-denumerability of the set of real numbers  --
3. Removing a denumerable set from a set of the power of the continuum  --
4. Sot of real numbers of an arbitrary interval  --
5. Sum of two sets of the power of the continuum  --
6. Cartesian product of a denumerable set and a set of the power of the continuum  --
7. Set of all infinite sequences of natural numbers  --
8. Cartesian product of two sets of the power of the continuum  --
9. Impossibility of a continuous (1-1) mapping of a plane on a straight line  --
10. Continuous curve filling up a square  --
11. Set of all infinite sequences of real numbers  --
12. Continuous curve filling up a denumerably dimensional cube  --
13. Set of all continuous functions  --
14. Decomposition of a set of natural numbers into a continuum of almost disjoint sets  --
Chapter V. Comparing the power of sets --
1. Sets of less and greater power  --
2. Sets of greater power than finite sets and denumerable sets. Hypothesis of the continuum  --
3. Cantor’s theorem on the set of all subsets of a given set  --
4. Generalized Continuum Hypothesis  --
5. Forming sets of ever greater powers  --
Chapter VI. Axiom of choice --
1. The axiom of choice. Controversy about it  --
2. The axiom of choice for a finite set of sets  --
2a. The axiom of choice for an infinite sequence of sets  --
3. Hilbert’s axiom  --
4. General principle of choice  --
5. Axiom of choice for finite sets  --
6. Examples of cases where we are able or not able to make an effective choice  --
7. Applications of the axiom of choice  --
8. The m-to-n correspondence  --
9. Dependent choices  --
Chapter VII. Cardinal numbers and operations on them --
1. Cardinal numbers  --
2. Sum of cardinal numbers  --
3. Product of two cardinal numbers  --
4. Exponentiation of cardinal numbers  --
5. Power of the set of all subsets of a given set  --
Chapter VIII. Inequalities for cardinal numbers --
1. Definition of an inequality between two cardinal numbers  --
2. Transitivity of the relation of inequality. Addition of inequalities  --
3. Exponentiation of inequalities for cardinal numbers  --
4. Relation m ≤ * n  --
Chapter IX. Difference of cardinal numbers --
1. Theorem of A. Tarski and F. Bernstein  --
2. Theorem on increasing the diminuend  --
3. Theorem on increasing the subtrahend  --
4. Difference in which the subtrahend is a natural number  --
5. Proof of formula 2m —m = 2m for m ≥ ϰ0 without the aid of the axiom of choice  --
6. Quotient of cardinal numbers  --
Chapter X. Infinite series and infinite products of cardinal numbers --
1. Sum of an infinite series of cardinal numbers  --
2. Properties of an infinite series of cardinal numbers  --
3. Examples of infinite series of cardinal numbers  --
4. Sum of an arbitrary set of cardinal numbers  --
5. Infinite product of cardinal numbers  --
6. Properties of infinite products of cardinal numbers. Examples  --
7. Theorem of J. König  --
8. Product of an arbitrary set of cardinal numbers  --
Chapter XI. Ordered sets --
1. Ordered sets  --
2. Partially ordered sets  --
3. Lattices  --
4. Similarity of sets  --
5. First and last element of an ordered set. Cuts. Jumps. Density and continuity of a set  --
6. Finite ordered sets  --
7. Sets of type ω  --
8. Sets of type ŋ  --
9. Dense ordered sets as subsets of continuous sets  --
10. Sets of type λ  --
Chapter XII. Order types and operations on them --
1. Order types  --
2. Sum of two ordered types  --
3. Product of two order types  --
4. Sum of an infinite series of order types  --
5. Power of the set of all denumerable order types  --
6. Power of the set of all order types of the power of the continuum  --
7. Sum of an arbitrary ordered set of order types  --
8. Infinite products of order types  --
9. Segments and remainders of order types  --
10. Divisors of order types  --
11. Comparison of order types  --
Chapter XIII. Well-ordered sets --
1. Well-ordered sets  --
2. The principle of transfinite induction  --
3. Induction for ordered sets  --
4. Similar mapping of a well-ordered set on its subset  --
5. Properties of segments of well-ordered sets  --
Chapter XIV. Ordinal numbers --
1. Ordinal numbers. Ordinal numbers as indices of the elements of well-ordered sets  --
2. Sets of ordinal numbers  --
3. Sum of ordinal numbers  --
4. Properties of the sum of ordinal numbers. Numbers of the. 1-st and of the 2-nd kind  --
5. Remainders of ordinal numbers  --
6. Prime components  --
7. Transfinite sequences of ordinal numbers and their limits  --
8. Infinite series of ordinal numbers and their sums  --
9. Product of ordinal numbers  --
10. Properties of the product of ordinal numbers  --
11. Theorem on the division of ordinal numbers  --
12. Divisors of ordinal numbers  --
13. Prime factors of ordinal numbers  --
14. Certain properties of prime components  --
15. Exponentiation of ordinal numbers  --
16. Definitions by transfinite induction  --
17. Transfinite products of ordinal numbers  --
18. Properties of the powers of ordinal numbers  --
19. The power ωa. Normal expansions of ordinal numbers  --
20. Epsilon numbers  --
21. Applications of the normal form  --
22. Determination of all cardinal numbers that are prime factors  --
23. Expanding ordinal numbers into prime factors  --
24. Roots of ordinal numbers  --
25. On ordinal numbers commutative with respect to addition  --
26. On ordinal numbers commutative with respect to multiplication  --
27. On the equation aB = Ba for ordinal numbers  --
28. Natural sum and natural product of ordinal numbers  --
29. Exponentiation of order types  --
Chapter XV. Number classes and alephs --
1. Numbers of the 1-st and of the 2-nd class  --
2. Cardinal number ϰ1  --
3. ϰ1— 2X° hypothesis  --
4. Properties of ordinal numbers of the 2-nd class  --
5. Transfinite induction for numbers of the 1-st class and of the 2-nd class  --
6. Convergence and limit of transfinite sequences of real numbers  --
7. Initial numbers, alephs and their notation  --
8. Formula ϰ2a = ϰa and conclusions from it  --
9. A proposition of elementary geometry, equivalent to the continuum hypothesis  --
10. Difference of alephs. Sums and products of transfinite sequences of successive alephs  --
11. Limit of transfinite sequences of initial numbers. Regular and singular initial numbers. Inaccessible alephs  --
Chapter XVI. Zermelo’s theorem and other theorems equivalent to the axiom of choice --
1. Equivalence of the axiom od choice, of Zermelo’s theorem and of the problem of trichotomy  --
2. Various theorems on cardinal numbers equivalent to the axiom of choice  --
3. A. Lindenbaum’s theorem  --
4. Equivalence of the axiom of choice to the theorems of Zorn and Teichmüller  --
5. Inference of the axiom of choice from the generalized continuum hypothesis  --
Chapter XVII. Applications of Zermelo’s theorem --
1. Hamel’s basis  --
2. Plane set having exactly two points in common with every straight line  --
3. Decomposition of an infinite set of power tn into more than tn almost disjoint sets  --
4. Some theorems on families of subsets of sets of given power  --
5. The power of the set of all order types of a given power  --
6. Applications of Zermelo’s theorem to the theory of ordered sets  --
Appendix  --
Bibliography  --
Index  --

MR, 20 #2288