Sets for MathematicsAdvanced undergraduate or beginning graduate students need a unified foundation for their study of geometry, analysis, and algebra. For the first time, this book uses categorical algebra to build such a foundation, starting from intuitive descriptions of mathematically and physically common phenomena and advancing to a precise specification of the nature of Categories of Sets. Set theory as the algebra of mappings is introduced and developed as a unifying basis for advanced mathematical subjects such as algebra, geometry, analysis, and combinatorics. The formal study evolves from general axioms that express universal properties of sums, products, mapping sets, and natural number recursion. |
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Inhalt
62 Truth Values for TwoStage Variable Sets | 114 |
63 Additional Exercises | 117 |
Consequences and Uses of Exponentials | 120 |
72 The Distributive Law | 126 |
73 Cantors Diagonal Argument | 129 |
74 Additional Exercises | 134 |
More on Power Sets | 136 |
82 The Covariant Power Set Functor | 141 |
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| 75 | |
| 78 | |
| 80 | |
| 84 | |
| 85 | |
45 Split Images | 89 |
46 The Axiom of Choice as the Distinguishing Property of ConstantRandom Sets | 92 |
47 Additional Exercises | 94 |
Mapping Sets and Exponentials | 96 |
52 Exponentiation | 98 |
53 Functoriality of Function Spaces | 102 |
54 Additional Exercises | 108 |
Summary of the Axioms and an Example of Variable Sets | 111 |
83 The Natural Map PX2²ˣ | 145 |
84 Measuring Averaging and Winning with VValued Quantities | 148 |
85 Additional Exercises | 152 |
Introduction to Variable Sets | 154 |
92 Recursion | 157 |
93 Arithmetic of N | 160 |
94 Additional Exercises | 165 |
Models of Additional Variation | 167 |
102 Actions | 171 |
103 Reversible Graphs | 176 |
104 Chaotic Graphs | 180 |
105 Feedback and Control | 186 |
106 To and from Idempotents | 189 |
107 Additional Exercises | 191 |
Logic as the Algebra of Parts | 193 |
A1 Basic Operators and Their Rules of Inference | 195 |
A2 Fields Nilpotents Idempotents | 212 |
The Axiom of Choice and Maximal Principles | 220 |
Definitions Symbols and the Greek Alphabet | 231 |
C2 Mathematical Notations and Logical Symbols | 251 |
C3 The Greek Alphabet | 252 |
Bibliography | 253 |
Index | 257 |
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Häufige Begriffe und Wortgruppen
abstract sets action Additional Exercises algebra arbitrary maps arrows axiom of choice binary Boolean called Cantor category of abstract category of sets characteristic function codomain cograph colimits commutative composition consider constant construction coproduct define Definition denoted determined diagonal diagram domain dual elements endomap epimorphism equal equations exactly example exists exponentiation fiber finite limits functor given graph hence Hint homomorphism idempotent implies inverse image isomorphism linear logic M-sets mapping sets mathematical Maximal Principle means monoid monomapping monomorphism morphisms natural transformation nilpotent notation operation parameterization partially ordered sets partition poset Proof Proposition prove pullback real numbers recursion retraction right 4-set ring rule of inference satisfies set theory sets and arbitrary sets and mappings Show slice category specific statement structure sups surjective terminal object theorem topos true truth values unique map universal mapping property variable sets vector space
Beliebte Passagen
Seite 1 - An abstract set is supposed to have elements, each of which has no structure, and is itself supposed to have no internal structure, except that the elements can be distinguished as equal or unequal, and to have no external structure except for the number of elements.
Seite 241 - The set 1 is characterized by the fact that for any set A there is exactly one mapping with domain A and codomain 1 . In symbols,
Seite 116 - X if and only if the following three conditions are all satisfied. A...
Seite 89 - X is called an equivalence relation iff it is reflexive, symmetric, and transitive.
Seite 231 - Let X and A be categories and F : X — *- A and G : A — *- X be functors.
Seite 3 - ... for each element a of A there is exactly one element b of B...
Verweise auf dieses Buch
From a Geometrical Point of View: A Study of the History and Philosophy of ... Jean-Pierre Marquis Eingeschränkte Leseprobe - 2008 |
The Architecture of Modern Mathematics:Essays in History and Philosophy ... J. Ferreiros,J. J. Gray Keine Leseprobe verfügbar - 2006 |
Bibliografische Informationen
| Titel | Sets for Mathematics |
| Autoren | F. William Lawvere, Robert Rosebrugh |
| Ausgabe | illustriert |
| Verlag | Cambridge University Press, 2003 |
| ISBN | 0521010608, 9780521010603 |
| Länge | 261 Seiten |
| Zitat exportieren | BiBTeX EndNote RefMan |