## 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 cmTema(s): Set theory | Transfinite numbers | Cardinal numbers | Numbers, OrdinalOtra clasificación: 03E10Chapter I. Sets and elementary set operations 1. Sets [5] 2. Elements of a set [5] 3. Symbols E and not E [5] 4. Set consisting of one element [5] 5. The empty set [6] 6. Equality of sets [6] 7. Sets of sets [7] 8. Subset of a set [8] 9. Sum of sets [9] 10. Difference of sets [10] 11. Product of sets [11] 12. Disjoint sums [13] 13. Complement of a set [15] 14. Ordered pairs [16] , 15. Correspondence. Function [16] 16. One-to-one correspondence [17] 17. Cartesian product of sets [19] 18. Exponentiation of sets [21] Chapter II. Equivalent sets 1. Equivalent sets. Relation ~ [22] 2. Finite and infinite sets [22] 3. Fundamental properties of the relation ~ [24] 4. Effectively equivalent sets [25] 5. Various theorems on the equivalence of sets [26] 6. The Cantor-Bernstein Theorem [30] Chapter III. Denumerable and non-denumerable sets 1. Denumerable and effectively denumerable sets [34] 2. Effective denumerability of the set of all rational numbers [36] 3. Effective denumerability of the infinite set of non-overlapping intervals [37] 4. Effective denumerability of the set of all finite sequences of rational numbers [39] 5. Effective denumerability of the set of all algebraic numbers [42] 6. Non-denumerable sets [43] 1. Properties of sets containing denumerable subsets [45] 8. Sets infinite in the sense of Dedekind [49] 9. Various definitions of finite sets [30] 10. Denumerability of the Cartesian product of two denumerable sets [52] 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 [53] 2. Non-denumerability of the set of real numbers [53] 3. Removing a denumerable set from a set of the power of the continuum [54] 4. Sot of real numbers of an arbitrary interval [56] 5. Sum of two sets of the power of the continuum [58] 6. Cartesian product of a denumerable set and a set of the power of the continuum [59] 7. Set of all infinite sequences of natural numbers [59] 8. Cartesian product of two sets of the power of the continuum [61] 9. Impossibility of a continuous (1-1) mapping of a plane on a straight line [66] 10. Continuous curve filling up a square [68] 11. Set of all infinite sequences of real numbers [69] 12. Continuous curve filling up a denumerably dimensional cube [73] 13. Set of all continuous functions [75] 14. Decomposition of a set of natural numbers into a continuum of almost disjoint sets [77] Chapter V. Comparing the power of sets 1. Sets of less and greater power [80] 2. Sets of greater power than finite sets and denumerable sets. Hypothesis of the continuum [82] 3. Cantor’s theorem on the set of all subsets of a given set [83] 4. Generalized Continuum Hypothesis [85] 5. Forming sets of ever greater powers [85] Chapter VI. Axiom of choice 1. The axiom of choice. Controversy about it [88] 2. The axiom of choice for a finite set of sets [92] 2a. The axiom of choice for an infinite sequence of sets [93] 3. Hilbert’s axiom [93] 4. General principle of choice [94] 5. Axiom of choice for finite sets [97] 6. Examples of cases where we are able or not able to make an effective choice [105] 7. Applications of the axiom of choice [109] 8. The m-to-n correspondence [126] 9. Dependent choices [129] Chapter VII. Cardinal numbers and operations on them 1. Cardinal numbers [132] 2. Sum of cardinal numbers [133] 3. Product of two cardinal numbers [135] 4. Exponentiation of cardinal numbers [138] 5. Power of the set of all subsets of a given set [140] Chapter VIII. Inequalities for cardinal numbers 1. Definition of an inequality between two cardinal numbers [144] 2. Transitivity of the relation of inequality. Addition of inequalities [147] 3. Exponentiation of inequalities for cardinal numbers [152] 4. Relation m ≤ * n [155] Chapter IX. Difference of cardinal numbers 1. Theorem of A. Tarski and F. Bernstein [158] 2. Theorem on increasing the diminuend [163] 3. Theorem on increasing the subtrahend [165] 4. Difference in which the subtrahend is a natural number [167] 5. Proof of formula 2m —m = 2m for m ≥ ϰ0 without the aid of the axiom of choice [168] 6. Quotient of cardinal numbers [170] Chapter X. Infinite series and infinite products of cardinal numbers 1. Sum of an infinite series of cardinal numbers [172] 2. Properties of an infinite series of cardinal numbers [174] 3. Examples of infinite series of cardinal numbers [175] 4. Sum of an arbitrary set of cardinal numbers [177] 5. Infinite product of cardinal numbers [179] 6. Properties of infinite products of cardinal numbers. Examples [179] 7. Theorem of J. König [181] 8. Product of an arbitrary set of cardinal numbers [182] Chapter XI. Ordered sets 1. Ordered sets [185] 2. Partially ordered sets [187] 3. Lattices [193] 4. Similarity of sets [197] 5. First and last element of an ordered set. Cuts. Jumps. Density and continuity of a set [200] 6. Finite ordered sets [204] 7. Sets of type ω [205] 8. Sets of type ŋ [209] 9. Dense ordered sets as subsets of continuous sets [214] 10. Sets of type λ [217] Chapter XII. Order types and operations on them 1. Order types [222] 2. Sum of two ordered types [223] 3. Product of two order types [229] 4. Sum of an infinite series of order types [236] 5. Power of the set of all denumerable order types [238] 6. Power of the set of all order types of the power of the continuum [240] 7. Sum of an arbitrary ordered set of order types [243] 8. Infinite products of order types [244] 9. Segments and remainders of order types [247] 10. Divisors of order types [250] 11. Comparison of order types [254] Chapter XIII. Well-ordered sets 1. Well-ordered sets [258] 2. The principle of transfinite induction [259] 3. Induction for ordered sets [261] 4. Similar mapping of a well-ordered set on its subset [264] 5. Properties of segments of well-ordered sets [265] Chapter XIV. Ordinal numbers 1. Ordinal numbers. Ordinal numbers as indices of the elements of well-ordered sets [269] 2. Sets of ordinal numbers [270] 3. Sum of ordinal numbers [271] 4. Properties of the sum of ordinal numbers. Numbers of the. 1-st and of the 2-nd kind [274] 5. Remainders of ordinal numbers [277] 6. Prime components [279] 7. Transfinite sequences of ordinal numbers and their limits [284] 8. Infinite series of ordinal numbers and their sums [287] 9. Product of ordinal numbers [290] 10. Properties of the product of ordinal numbers [292] 11. Theorem on the division of ordinal numbers [294] 12. Divisors of ordinal numbers [298] 13. Prime factors of ordinal numbers [303] 14. Certain properties of prime components [305] 15. Exponentiation of ordinal numbers [306] 16. Definitions by transfinite induction [311] 17. Transfinite products of ordinal numbers [313] 18. Properties of the powers of ordinal numbers [315] 19. The power ωa. Normal expansions of ordinal numbers [319] 20. Epsilon numbers [323] 21. Applications of the normal form [327] 22. Determination of all cardinal numbers that are prime factors [332] 23. Expanding ordinal numbers into prime factors [340] 24. Roots of ordinal numbers [341] 25. On ordinal numbers commutative with respect to addition [343] 26. On ordinal numbers commutative with respect to multiplication [347] 27. On the equation aB = Ba for ordinal numbers [359] 28. Natural sum and natural product of ordinal numbers [363] 29. Exponentiation of order types [364] Chapter XV. Number classes and alephs 1. Numbers of the 1-st and of the 2-nd class [366] 2. Cardinal number ϰ1 [369] 3. ϰ1— 2X° hypothesis [376] 4. Properties of ordinal numbers of the 2-nd class [379] 5. Transfinite induction for numbers of the 1-st class and of the 2-nd class [386] 6. Convergence and limit of transfinite sequences of real numbers [387] 7. Initial numbers, alephs and their notation [389] 8. Formula ϰ2a = ϰa and conclusions from it [392] 9. A proposition of elementary geometry, equivalent to the continuum hypothesis [397] 10. Difference of alephs. Sums and products of transfinite sequences of successive alephs [399] 11. Limit of transfinite sequences of initial numbers. Regular and singular initial numbers. Inaccessible alephs [402] 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 [407] 2. Various theorems on cardinal numbers equivalent to the axiom of choice [414] 3. A. Lindenbaum’s theorem [426] 4. Equivalence of the axiom of choice to the theorems of Zorn and Teichmüller [427] 5. Inference of the axiom of choice from the generalized continuum hypothesis [434] Chapter XVII. Applications of Zermelo’s theorem 1. Hamel’s basis [440] 2. Plane set having exactly two points in common with every straight line [446] 3. Decomposition of an infinite set of power tn into more than tn almost disjoint sets [448] 4. Some theorems on families of subsets of sets of given power [450] 5. The power of the set of all order types of a given power [457] 6. Applications of Zermelo’s theorem to the theory of ordered sets [458] Appendix [468] Bibliography [469] Index [482]

Item type | Home library | Call number | Materials specified | Status | Date due | Barcode | Course reserves |
---|---|---|---|---|---|---|---|

Libros | Instituto de Matemática, CONICET-UNS | 03 Si572c (Browse shelf) | Available | A-2512 |

Bibliografía: p. [469]-481.

Chapter I. Sets and elementary set operations --

1. Sets [5] --

2. Elements of a set [5] --

3. Symbols E and not E [5] --

4. Set consisting of one element [5] --

5. The empty set [6] --

6. Equality of sets [6] --

7. Sets of sets [7] --

8. Subset of a set [8] --

9. Sum of sets [9] --

10. Difference of sets [10] --

11. Product of sets [11] --

12. Disjoint sums [13] --

13. Complement of a set [15] --

14. Ordered pairs [16] --

, 15. Correspondence. Function [16] --

16. One-to-one correspondence [17] --

17. Cartesian product of sets [19] --

18. Exponentiation of sets [21] --

Chapter II. Equivalent sets --

1. Equivalent sets. Relation ~ [22] --

2. Finite and infinite sets [22] --

3. Fundamental properties of the relation ~ [24] --

4. Effectively equivalent sets [25] --

5. Various theorems on the equivalence of sets [26] --

6. The Cantor-Bernstein Theorem [30] --

Chapter III. Denumerable and non-denumerable sets --

1. Denumerable and effectively denumerable sets [34] --

2. Effective denumerability of the set of all rational numbers [36] --

3. Effective denumerability of the infinite set of non-overlapping intervals [37] --

4. Effective denumerability of the set of all finite sequences of rational numbers [39] --

5. Effective denumerability of the set of all algebraic numbers [42] --

6. Non-denumerable sets [43] --

1. Properties of sets containing denumerable subsets [45] --

8. Sets infinite in the sense of Dedekind [49] --

9. Various definitions of finite sets [30] --

10. Denumerability of the Cartesian product of two denumerable sets [52] --

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 [53] --

2. Non-denumerability of the set of real numbers [53] --

3. Removing a denumerable set from a set of the power of the continuum [54] --

4. Sot of real numbers of an arbitrary interval [56] --

5. Sum of two sets of the power of the continuum [58] --

6. Cartesian product of a denumerable set and a set of the power of the continuum [59] --

7. Set of all infinite sequences of natural numbers [59] --

8. Cartesian product of two sets of the power of the continuum [61] --

9. Impossibility of a continuous (1-1) mapping of a plane on a straight line [66] --

10. Continuous curve filling up a square [68] --

11. Set of all infinite sequences of real numbers [69] --

12. Continuous curve filling up a denumerably dimensional cube [73] --

13. Set of all continuous functions [75] --

14. Decomposition of a set of natural numbers into a continuum of almost disjoint sets [77] --

Chapter V. Comparing the power of sets --

1. Sets of less and greater power [80] --

2. Sets of greater power than finite sets and denumerable sets. Hypothesis of the continuum [82] --

3. Cantor’s theorem on the set of all subsets of a given set [83] --

4. Generalized Continuum Hypothesis [85] --

5. Forming sets of ever greater powers [85] --

Chapter VI. Axiom of choice --

1. The axiom of choice. Controversy about it [88] --

2. The axiom of choice for a finite set of sets [92] --

2a. The axiom of choice for an infinite sequence of sets [93] --

3. Hilbert’s axiom [93] --

4. General principle of choice [94] --

5. Axiom of choice for finite sets [97] --

6. Examples of cases where we are able or not able to make an effective choice [105] --

7. Applications of the axiom of choice [109] --

8. The m-to-n correspondence [126] --

9. Dependent choices [129] --

Chapter VII. Cardinal numbers and operations on them --

1. Cardinal numbers [132] --

2. Sum of cardinal numbers [133] --

3. Product of two cardinal numbers [135] --

4. Exponentiation of cardinal numbers [138] --

5. Power of the set of all subsets of a given set [140] --

Chapter VIII. Inequalities for cardinal numbers --

1. Definition of an inequality between two cardinal numbers [144] --

2. Transitivity of the relation of inequality. Addition of inequalities [147] --

3. Exponentiation of inequalities for cardinal numbers [152] --

4. Relation m ≤ * n [155] --

Chapter IX. Difference of cardinal numbers --

1. Theorem of A. Tarski and F. Bernstein [158] --

2. Theorem on increasing the diminuend [163] --

3. Theorem on increasing the subtrahend [165] --

4. Difference in which the subtrahend is a natural number [167] --

5. Proof of formula 2m —m = 2m for m ≥ ϰ0 without the aid of the axiom of choice [168] --

6. Quotient of cardinal numbers [170] --

Chapter X. Infinite series and infinite products of cardinal numbers --

1. Sum of an infinite series of cardinal numbers [172] --

2. Properties of an infinite series of cardinal numbers [174] --

3. Examples of infinite series of cardinal numbers [175] --

4. Sum of an arbitrary set of cardinal numbers [177] --

5. Infinite product of cardinal numbers [179] --

6. Properties of infinite products of cardinal numbers. Examples [179] --

7. Theorem of J. König [181] --

8. Product of an arbitrary set of cardinal numbers [182] --

Chapter XI. Ordered sets --

1. Ordered sets [185] --

2. Partially ordered sets [187] --

3. Lattices [193] --

4. Similarity of sets [197] --

5. First and last element of an ordered set. Cuts. Jumps. Density and continuity of a set [200] --

6. Finite ordered sets [204] --

7. Sets of type ω [205] --

8. Sets of type ŋ [209] --

9. Dense ordered sets as subsets of continuous sets [214] --

10. Sets of type λ [217] --

Chapter XII. Order types and operations on them --

1. Order types [222] --

2. Sum of two ordered types [223] --

3. Product of two order types [229] --

4. Sum of an infinite series of order types [236] --

5. Power of the set of all denumerable order types [238] --

6. Power of the set of all order types of the power of the continuum [240] --

7. Sum of an arbitrary ordered set of order types [243] --

8. Infinite products of order types [244] --

9. Segments and remainders of order types [247] --

10. Divisors of order types [250] --

11. Comparison of order types [254] --

Chapter XIII. Well-ordered sets --

1. Well-ordered sets [258] --

2. The principle of transfinite induction [259] --

3. Induction for ordered sets [261] --

4. Similar mapping of a well-ordered set on its subset [264] --

5. Properties of segments of well-ordered sets [265] --

Chapter XIV. Ordinal numbers --

1. Ordinal numbers. Ordinal numbers as indices of the elements of well-ordered sets [269] --

2. Sets of ordinal numbers [270] --

3. Sum of ordinal numbers [271] --

4. Properties of the sum of ordinal numbers. Numbers of the. 1-st and of the 2-nd kind [274] --

5. Remainders of ordinal numbers [277] --

6. Prime components [279] --

7. Transfinite sequences of ordinal numbers and their limits [284] --

8. Infinite series of ordinal numbers and their sums [287] --

9. Product of ordinal numbers [290] --

10. Properties of the product of ordinal numbers [292] --

11. Theorem on the division of ordinal numbers [294] --

12. Divisors of ordinal numbers [298] --

13. Prime factors of ordinal numbers [303] --

14. Certain properties of prime components [305] --

15. Exponentiation of ordinal numbers [306] --

16. Definitions by transfinite induction [311] --

17. Transfinite products of ordinal numbers [313] --

18. Properties of the powers of ordinal numbers [315] --

19. The power ωa. Normal expansions of ordinal numbers [319] --

20. Epsilon numbers [323] --

21. Applications of the normal form [327] --

22. Determination of all cardinal numbers that are prime factors [332] --

23. Expanding ordinal numbers into prime factors [340] --

24. Roots of ordinal numbers [341] --

25. On ordinal numbers commutative with respect to addition [343] --

26. On ordinal numbers commutative with respect to multiplication [347] --

27. On the equation aB = Ba for ordinal numbers [359] --

28. Natural sum and natural product of ordinal numbers [363] --

29. Exponentiation of order types [364] --

Chapter XV. Number classes and alephs --

1. Numbers of the 1-st and of the 2-nd class [366] --

2. Cardinal number ϰ1 [369] --

3. ϰ1— 2X° hypothesis [376] --

4. Properties of ordinal numbers of the 2-nd class [379] --

5. Transfinite induction for numbers of the 1-st class and of the 2-nd class [386] --

6. Convergence and limit of transfinite sequences of real numbers [387] --

7. Initial numbers, alephs and their notation [389] --

8. Formula ϰ2a = ϰa and conclusions from it [392] --

9. A proposition of elementary geometry, equivalent to the continuum hypothesis [397] --

10. Difference of alephs. Sums and products of transfinite sequences of successive alephs [399] --

11. Limit of transfinite sequences of initial numbers. Regular and singular initial numbers. Inaccessible alephs [402] --

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 [407] --

2. Various theorems on cardinal numbers equivalent to the axiom of choice [414] --

3. A. Lindenbaum’s theorem [426] --

4. Equivalence of the axiom of choice to the theorems of Zorn and Teichmüller [427] --

5. Inference of the axiom of choice from the generalized continuum hypothesis [434] --

Chapter XVII. Applications of Zermelo’s theorem --

1. Hamel’s basis [440] --

2. Plane set having exactly two points in common with every straight line [446] --

3. Decomposition of an infinite set of power tn into more than tn almost disjoint sets [448] --

4. Some theorems on families of subsets of sets of given power [450] --

5. The power of the set of all order types of a given power [457] --

6. Applications of Zermelo’s theorem to the theory of ordered sets [458] --

Appendix [468] --

Bibliography [469] --

Index [482] --

MR, 20 #2288

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