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## Cardinal and ordinal numbers / Waclaw Sierpinski.

Idioma: Inglés Lenguaje original: Polaco Editor: Warszawa : PWN, 1965Edición: 2nd ed., rev. / [translated from Polish by Janina Smólska]Descripción: 491 p. ; 25 cmOtra clasificación: 03E10
Contenidos:
``` CONTENTS
Foreword [7]
Chapter I. Sets and elementary set operations
1. Sets [9]
2. Elements of a set [9]
3. Symbols E and not E [9]
4. Set consisting of one element [9]
5. The empty set [10]
6. Equality of sets [10]
7. Sets of sets [10]
8. Subset of a set [11]
9. Sum of sets [13]
10. Difference of sets [14]
11. Product of sets [15]
12. Disjoint sums [17]
13. Complement of a set [19]
14. Ordered pairs [20]
15. Correspondence. Function [20]
16. One-to-one correspondence [21]
17. Cartesian product of sets [22]
18. Exponentiation of sets [24]
Chapter II. Equivalent sets
1. Equivalent sets. Relation ~ [26]
2. Finite and infinite sets [26]
3. Fundamental properties of the relation ~ [28]
4. Effectively equivalent sets [29]
5. Various theorems on the equivalence of sets [30]
6. The Cantor-Bernstein Theorem [34]
Chapter III. Denumerable and non-denumerable sets
1. Denumerable and effectively denumerable sets [38]
2. Effective denumerability of the set of all rational numbers [40]
3. Effective denumerability of the infinite set of non-overlapping intervals [41]
4. Effective denumerability of the set of all finite sequences of rational numbers [43]
5. Effective denumerability of the set of all algebraic numbers [46]
6. Non-denumerable sets [47]
7. Properties of sets containing denumerable subsets [49]
8. Sets infinite in the sense of Dedekind [53]
9. Various definitions of finite sets [54]
10. Denumerability of the Cartesian product of two denumerable sets [56]
Chapter IV. Sets of the power of the continuum
1. Sets of the power of the continuum and sets effectively of the power of the continuum [57]
2. Non-denumerability of the set of real numbers [57]
3. Removing a denumerable set from a set of the power of the continuum [58]
4. Set of real numbers of an arbitrary interval [60]
5. Sum of two sets of the power of the continuum [62]
6. Cartesian product of a denumerable set and a set of the power of the continuum [63]
7. Set of all infinite sequences of natural numbers [63]
8. Cartesian product of two sets of the power of the continuum [65]
9. Impossibility of a continuous 1-1 mapping of a plane on a straight line [70]
10. Continuous curve filling up a square [71]
11. Set of all infinite sequences of real numbers [73]
12. Continuous curve filling up a denumerably dimensional cube [77]
13. Set of all continuous functions [79]
14. Decomposition of a set of natural numbers into a continuum of almost disjoint sets [81]
Chapter V. Comparing the power of sets
1. Sets of different power [84]
2. Sets of greater power than finite sets and denumerable sets. Hypothesis of the continuum [86]
3. Cantor’s theorem on the set of all subsets of a given set [87]
4. Generalized Continuum Hypothesis [88]
5. Forming sets of ever greater powers [89]
Chapter VI. Axiom of choice
1. The axiom of choice. Controversy about it [92]
2. The axiom of choioe for a finite Bet of sets [96]
2a. The axiom of choice for an infinite sequence of sets [97]
3. Hilbert’s axiom [97]
4. General principle of choice [98]
5. Axiom of choice for finite sets [101]
6. Examples of cases where we are able and where we are not able to make an effective choice [109]
7. Applications of the axiom of choice [112]
8. The m-to-n correspondence [129]
9. Dependent choices [132]
Chapter VII. Cardinal numbers and operations on them
1. Cardinal numbers [135]
2. Sum of cardinal numbers [136]
3. Product of two cardinal numbers [138]
4. Exponentiation of cardinal numbers [141]
5. Power of the set of all subsets of a given set [143]
Chapter VIII. Inequalities for cardinal numbers
1. Definition of an inequality between two cardinal numbers [147]
2. Transitivity of the relation of inequality Addition of inequalities [150]
3. Exponentiation of inequalities for cardinal numbers [155]
4. Relation m < * n [157]
Chapter IX. Difference of cardinal numbers
1. Theorem of A. Tarski and F. Bernstein [161]
2. Theorem on increasing the diminuend [166]
3. Theorem on increasing the subtrahend [168]
4. Difference in which the subtrahend is a natural number [170]
5. Proof of the formula 2m—m = 2m for m > No without the aid of the axiom of choice [172]
6. Quotient of cardinal numbers [173]
Chapter X. Infinite series and infinite products of cardinal numbers
1. Sum of an infinite series of cardinal numbers [175]
2. Properties of an infinite series of cardinal numbers [177]
3. Examples of infinite series of cardinal numbers [178]
4. Sum of an arbitrary set of cardinal numbers [180]
5. Infinite product of cardinal numbers [182]
6. Properties of infinite products of cardinal numbers. Examples [183]
7. Theorem of J. König [184]
8. Product of an arbitrary set of cardinal numbers [185]
Chapter XI. Ordered sets
1. Ordered sets [188]
2. Partially ordered sets [190]
3. Lattices [196]
4. Similarity of sets [200]
5. First and last element of an ordered set. Cuts. Jumps. Density and continuity of a set [203]
6. Finite ordered sets [207]
7. Sets of type ω [208]
8. Sets of type ŋ 97 [212]
9. Dense ordered sets as subsets of continuous sets [217]
10. Sets of type λ [220]
Chapter XII. Order types and operations on them
1. Order types [225]
2. Sum of two order types [226]
3. Product of two order types [232]
4. Sum of an infinite series of order, types [239]
5. Power of the set of all denumerable order types [242]
6. Power of the set of all order types of the power of the continuum [243]
7. Sum of an arbitrary ordered set of order types [246]
8. Infinite products of order types [247]
9. Segments and remainders of order types [250]
10. Divisors of order types [254]
11. Comparison of order types [257]
Chapter XIII. Well-ordered sets
1. Well-ordered sets [261]
2. The principle of transfinite induction [262]
3. Induction for ordered sets [264]
4. Similar mapping of a well-ordered set on its subset [267]
5. Properties of segments of well-ordered sets. Principal theorem on wellordered sets [268]
Chapter XIV. Ordinal numbers
1. Ordinal numbers. Ordinal numbers as indices of the elements of wellordered sets [272]
2. Sets of ordinal numbers [273]
3. Sum of ordinal numbers [274]
4. Properties of the sum of ordinal numbers. Numbers of the 1-st and of the 2-nd kind [277]
5. Remainders of ordinal numbers [280]
6. Prime components [282]
7. Transfinite sequences of ordinal numbers and their limits [287]
8. Infinite series of ordinal numbers and their sums [290]
9. Product of ordinal numbers [293]
10. Properties of the product of ordinal numbers [295]
11. Theorem on the division of ordinal numbers [297]
12. Divisors of ordinal numbers [301]
13. Prime factors of ordinal numbers [306]
14. Certain properties of prime components [308]
15. Exponentiation of ordinal numbers [309]
16. Definitions by transfinite induction [314]
17.. Transfinite products of ordinal numbers [316]
18. Properties of the powers of ordinal numbers [318]
19. The power ωa. Normal expansions of ordinal numbers [322]
20. Epsilon numbers [326]
21. Applications of the normal form
22. Determination of all ordinal numbers that are prime factors [335]
23. Expanding ordinal numbers into prime factors [343]
24. Roots of ordinal numbers [345]
25. On ordinal numbers commutative with respect to addition [340]
26. On ordinal numbers commutative with respect to multiplication [351]
27. On the equation aß = ßa for ordinal numbers [333]
28. Natural sum and natural product of ordinal numbers [333]
29. Exponentiation of order types [337]
Chapter XV. Number classes and alephs
1. Numbers of the 1-st and of the 2-nd class [339]
2. Cardinal number [372]
3. ϰ1= 2ϰ0 hypothesis [378]
4. Properties of ordinal numbers of the 2-nd class [382]
5. Transfinite induction for numbers of the 1-st class and of the 2-nd class [389]
6. Convergence and limit of transfinite sequences of real numbers [390]
7. Initial numbers, alephs and their notation [391]
8. Formula ϰ2a = ϰa and conclusions from it [395]
9. A proposition of elementary geometry, equivalent to the continuum hypothesis [400]
10. Difference of alephs. Sums and products of transfinite sequences of successive alephs [402]
11. Limit of transfinite sequences of initial numbers. Regular and singular initial numbers. Inaccessible alephs [405]
Chapter XVI. Zermelo’s theorem and other theorems equivalent to the axiom of choice
1. Equivalence of the axiom of choice, of Zermelo’s theorem and of the problem of trichotomy [410]
2. Various theorems on cardinal numbers equivalent to the axiom of choice [417]
3. A. Lindenbaum’s theorem [429]
4. Equivalence of the axiom of choice to the theorems of Zorn and Teichmüfiller [430]
5. Inference of the axiom of choice from the generalized continuum hypothesis [437]
Chapter XVII. Applications of Zermelo’s theorem
1. Hamel’s basis [443]
2. Plane set having exactly two points in common with every straight line [449]
3. Decomposition of an infinite set of power m into more than m almost disjoint sets [451]
4. Some theorems on families of subsets of sets of given powers [453]
5. The power of the set of all order types of a given power [459]
6. Applications of Zermelo’s theorem to the theory of ordered sets [461]
Appendix [470]
Bibliography [471]
Index [485]
```
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CONTENTS --
Foreword [7] --
Chapter I. Sets and elementary set operations --
1. Sets [9] --
2. Elements of a set [9] --
3. Symbols E and not E [9] --
4. Set consisting of one element [9] --
5. The empty set [10] --
6. Equality of sets [10] --
7. Sets of sets [10] --
8. Subset of a set [11] --
9. Sum of sets [13] --
10. Difference of sets [14] --
11. Product of sets [15] --
12. Disjoint sums [17] --
13. Complement of a set [19] --
14. Ordered pairs [20] --
15. Correspondence. Function [20] --
16. One-to-one correspondence [21] --
17. Cartesian product of sets [22] --
18. Exponentiation of sets [24] --
Chapter II. Equivalent sets --
1. Equivalent sets. Relation ~ [26] --
2. Finite and infinite sets [26] --
3. Fundamental properties of the relation ~ [28] --
4. Effectively equivalent sets [29] --
5. Various theorems on the equivalence of sets [30] --
6. The Cantor-Bernstein Theorem [34] --
Chapter III. Denumerable and non-denumerable sets --
1. Denumerable and effectively denumerable sets [38] --
2. Effective denumerability of the set of all rational numbers [40] --
3. Effective denumerability of the infinite set of non-overlapping intervals [41] --
4. Effective denumerability of the set of all finite sequences of rational numbers [43] --
5. Effective denumerability of the set of all algebraic numbers [46] --
6. Non-denumerable sets [47] --
7. Properties of sets containing denumerable subsets [49] --
8. Sets infinite in the sense of Dedekind [53] --
9. Various definitions of finite sets [54] --
10. Denumerability of the Cartesian product of two denumerable sets [56] --
Chapter IV. Sets of the power of the continuum --
1. Sets of the power of the continuum and sets effectively of the power of the continuum [57] --
2. Non-denumerability of the set of real numbers [57] --
3. Removing a denumerable set from a set of the power of the continuum [58] --
4. Set of real numbers of an arbitrary interval [60] --
5. Sum of two sets of the power of the continuum [62] --
6. Cartesian product of a denumerable set and a set of the power of the continuum [63] --
7. Set of all infinite sequences of natural numbers [63] --
8. Cartesian product of two sets of the power of the continuum [65] --
9. Impossibility of a continuous 1-1 mapping of a plane on a straight line [70] --
10. Continuous curve filling up a square [71] --
11. Set of all infinite sequences of real numbers [73] --
12. Continuous curve filling up a denumerably dimensional cube [77] --
13. Set of all continuous functions [79] --
14. Decomposition of a set of natural numbers into a continuum of almost disjoint sets [81] --
Chapter V. Comparing the power of sets --
1. Sets of different power [84] --
2. Sets of greater power than finite sets and denumerable sets. Hypothesis of the continuum [86] --
3. Cantor’s theorem on the set of all subsets of a given set [87] --
4. Generalized Continuum Hypothesis [88] --
5. Forming sets of ever greater powers [89] --
Chapter VI. Axiom of choice --
1. The axiom of choice. Controversy about it [92] --
2. The axiom of choioe for a finite Bet of sets [96] --
2a. The axiom of choice for an infinite sequence of sets [97] --
3. Hilbert’s axiom [97] --
4. General principle of choice [98] --
5. Axiom of choice for finite sets [101] --
6. Examples of cases where we are able and where we are not able to make an effective choice [109] --
7. Applications of the axiom of choice [112] --
8. The m-to-n correspondence [129] --
9. Dependent choices [132] --
Chapter VII. Cardinal numbers and operations on them --
1. Cardinal numbers [135] --
2. Sum of cardinal numbers [136] --
3. Product of two cardinal numbers [138] --
4. Exponentiation of cardinal numbers [141] --
5. Power of the set of all subsets of a given set [143] --
Chapter VIII. Inequalities for cardinal numbers --
1. Definition of an inequality between two cardinal numbers [147] --
2. Transitivity of the relation of inequality Addition of inequalities [150] --
3. Exponentiation of inequalities for cardinal numbers [155] --
4. Relation m < * n [157] --
Chapter IX. Difference of cardinal numbers --
1. Theorem of A. Tarski and F. Bernstein [161] --
2. Theorem on increasing the diminuend [166] --
3. Theorem on increasing the subtrahend [168] --
4. Difference in which the subtrahend is a natural number [170] --
5. Proof of the formula 2m—m = 2m for m > No without the aid of the axiom of choice [172] --
6. Quotient of cardinal numbers [173] --
Chapter X. Infinite series and infinite products of cardinal numbers --
1. Sum of an infinite series of cardinal numbers [175] --
2. Properties of an infinite series of cardinal numbers [177] --
3. Examples of infinite series of cardinal numbers [178] --
4. Sum of an arbitrary set of cardinal numbers [180] --
5. Infinite product of cardinal numbers [182] --
6. Properties of infinite products of cardinal numbers. Examples [183] --
7. Theorem of J. König [184] --
8. Product of an arbitrary set of cardinal numbers [185] --
Chapter XI. Ordered sets --
1. Ordered sets [188] --
2. Partially ordered sets [190] --
3. Lattices [196] --
4. Similarity of sets [200] --
5. First and last element of an ordered set. Cuts. Jumps. Density and continuity of a set [203] --
6. Finite ordered sets [207] --
7. Sets of type ω [208] --
8. Sets of type ŋ 97 [212] --
9. Dense ordered sets as subsets of continuous sets [217] --
10. Sets of type λ [220] --
Chapter XII. Order types and operations on them --
1. Order types [225] --
2. Sum of two order types [226] --
3. Product of two order types [232] --
4. Sum of an infinite series of order, types [239] --
5. Power of the set of all denumerable order types [242] --
6. Power of the set of all order types of the power of the continuum [243] --
7. Sum of an arbitrary ordered set of order types [246] --
8. Infinite products of order types [247] --
9. Segments and remainders of order types [250] --
10. Divisors of order types [254] --
11. Comparison of order types [257] --
Chapter XIII. Well-ordered sets --
1. Well-ordered sets [261] --
2. The principle of transfinite induction [262] --
3. Induction for ordered sets [264] --
4. Similar mapping of a well-ordered set on its subset [267] --
5. Properties of segments of well-ordered sets. Principal theorem on wellordered sets [268] --
Chapter XIV. Ordinal numbers --
1. Ordinal numbers. Ordinal numbers as indices of the elements of wellordered sets [272] --
2. Sets of ordinal numbers [273] --
3. Sum of ordinal numbers [274] --
4. Properties of the sum of ordinal numbers. Numbers of the 1-st and of the 2-nd kind [277] --
5. Remainders of ordinal numbers [280] --
6. Prime components [282] --
7. Transfinite sequences of ordinal numbers and their limits [287] --
8. Infinite series of ordinal numbers and their sums [290] --
9. Product of ordinal numbers [293] --
10. Properties of the product of ordinal numbers [295] --
11. Theorem on the division of ordinal numbers [297] --
12. Divisors of ordinal numbers [301] --
13. Prime factors of ordinal numbers [306] --
14. Certain properties of prime components [308] --
15. Exponentiation of ordinal numbers [309] --
16. Definitions by transfinite induction [314] --
17.. Transfinite products of ordinal numbers [316] --
18. Properties of the powers of ordinal numbers [318] --
19. The power ωa. Normal expansions of ordinal numbers [322] --
20. Epsilon numbers [326] --
21. Applications of the normal form --
22. Determination of all ordinal numbers that are prime factors [335] --
23. Expanding ordinal numbers into prime factors [343] --
24. Roots of ordinal numbers [345] --
25. On ordinal numbers commutative with respect to addition [340] --
26. On ordinal numbers commutative with respect to multiplication [351] --
27. On the equation aß = ßa for ordinal numbers [333] --
28. Natural sum and natural product of ordinal numbers [333] --
29. Exponentiation of order types [337] --
Chapter XV. Number classes and alephs --
1. Numbers of the 1-st and of the 2-nd class [339] --
2. Cardinal number [372] --
3. ϰ1= 2ϰ0 hypothesis [378] --
4. Properties of ordinal numbers of the 2-nd class [382] --
5. Transfinite induction for numbers of the 1-st class and of the 2-nd class [389] --
6. Convergence and limit of transfinite sequences of real numbers [390] --
7. Initial numbers, alephs and their notation [391] --
8. Formula ϰ2a = ϰa and conclusions from it [395] --
9. A proposition of elementary geometry, equivalent to the continuum hypothesis [400] --
10. Difference of alephs. Sums and products of transfinite sequences of successive alephs [402] --
11. Limit of transfinite sequences of initial numbers. Regular and singular initial numbers. Inaccessible alephs [405] --
Chapter XVI. Zermelo’s theorem and other theorems equivalent to the axiom of choice --
1. Equivalence of the axiom of choice, of Zermelo’s theorem and of the problem of trichotomy [410] --
2. Various theorems on cardinal numbers equivalent to the axiom of choice [417] --
3. A. Lindenbaum’s theorem [429] --
4. Equivalence of the axiom of choice to the theorems of Zorn and Teichmüfiller [430] --
5. Inference of the axiom of choice from the generalized continuum hypothesis [437] --
Chapter XVII. Applications of Zermelo’s theorem --
1. Hamel’s basis [443] --
2. Plane set having exactly two points in common with every straight line [449] --
3. Decomposition of an infinite set of power m into more than m almost disjoint sets [451] --
4. Some theorems on families of subsets of sets of given powers [453] --
5. The power of the set of all order types of a given power [459] --
6. Applications of Zermelo’s theorem to the theory of ordered sets [461] --
Appendix [470] --
Bibliography [471] --
Index [485] --

MR, 33 #2549

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