Set Theory With an Introduction to Real Point Sets /
What is a number? What is infinity? What is continuity? What is order? Answers to these fundamental questions obtained by late nineteenth-century mathematicians such as Dedekind and Cantor gave birth to set theory. This textbook presents classical set theory in an intuitive but concrete manner. To a...
| Κύριος συγγραφέας: | |
|---|---|
| Συγγραφή απο Οργανισμό/Αρχή: | |
| Μορφή: | Ηλεκτρονική πηγή Ηλ. βιβλίο |
| Γλώσσα: | English |
| Έκδοση: |
New York, NY :
Springer New York : Imprint: Birkhäuser,
2014.
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| Θέματα: | |
| Διαθέσιμο Online: | Full Text via HEAL-Link |
Πίνακας περιεχομένων:
- 1 Preliminaries: Sets, Relations, and Functions
- Part I Dedekind: Numbers
- 2 The Dedekind–Peano Axioms
- 3 Dedekind’s Theory of the Continuum
- 4 Postscript I: What Exactly Are the Natural Numbers?
- Part II Cantor: Cardinals, Order, and Ordinals
- 5 Cardinals: Finite, Countable, and Uncountable
- 6 Cardinal Arithmetic and the Cantor Set
- 7 Orders and Order Types
- 8 Dense and Complete Orders
- 9 Well-Orders and Ordinals
- 10 Alephs, Cofinality, and the Axiom of Choice
- 11 Posets, Zorn’s Lemma, Ranks, and Trees
- 12 Postscript II: Infinitary Combinatorics
- Part III Real Point Sets
- 13 Interval Trees and Generalized Cantor Sets
- 14 Real Sets and Functions
- 15 The Heine–Borel and Baire Category Theorems
- 16 Cantor–Bendixson Analysis of Countable Closed Sets
- 17 Brouwer’s Theorem and Sierpinski’s Theorem
- 18 Borel and Analytic Sets
- 19 Postscript III: Measurability and Projective Sets
- Part IV Paradoxes and Axioms
- 20 Paradoxes and Resolutions
- 21 Zermelo–Fraenkel System and von Neumann Ordinals
- 22 Postscript IV: Landmarks of Modern Set Theory
- Appendices
- A Proofs of Uncountability of the Reals
- B Existence of Lebesgue Measure
- C List of ZF Axioms
- References
- List of Symbols and Notations
- Index.
