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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 [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]```
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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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