Numbers, Sets and Axioms: The Apparatus of MathematicsFollowing the success of Logic for Mathematicians, Dr Hamilton has written a text for mathematicians and students of mathematics that contains a description and discussion of the fundamental conceptual and formal apparatus upon which modern pure mathematics relies. The author's intention is to remove some of the mystery that surrounds the foundations of mathematics. He emphasises the intuitive basis of mathematics; the basic notions are numbers and sets and they are considered both informally and formally. The role of axiom systems is part of the discussion but their limitations are pointed out. Formal set theory has its place in the book but Dr Hamilton recognises that this is a part of mathematics and not the basis on which it rests. Throughout, the abstract ideas are liberally illustrated by examples so this account should be well-suited, both specifically as a course text and, more broadly, as background reading. The reader is presumed to have some mathematical experience but no knowledge of mathematical logic is required. |
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Contents
Numbers | 1 |
12 Rational numbers | 18 |
13 Real numbers | 26 |
14 Decimal notation | 43 |
The size of a set | 51 |
22 Uncountable sets | 63 |
23 Cardinal numbers | 73 |
Ordered sets | 82 |
45 Models of set theory | 156 |
The axiom of choice | 163 |
52 Zorns lemma and the wellordering theorem | 171 |
53 Other consequences of the axiom of choice | 184 |
Ordinal and cardinal numbers | 192 |
62 Transfinite recursion and ordinal arithmetic | 205 |
63 Cardinal numbers | 221 |
Hints and solutions to selected exercises | 237 |
32 Lattices and Boolean algebras | 97 |
Set theory | 108 |
42 The ZermeloFraenkel axioms | 115 |
43 Mathematics in ZF | 129 |
44 Sets and classes | 145 |
| 247 | |
| 249 | |
| 252 | |
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Common terms and phrases
addition and multiplication aleph axiom of choice bijection Boolean algebra cardinal number Cartesian product Cauchy sequence Chapter cofinal collection consequence construction contradiction Corollary countable sets define definition denote disjoint equinumerous equivalence classes equivalence relation example exercise exist finite set formal generalised Given any set given set greatest element Hence implies infinite set initial segment injection intuitive lattice least element least upper bound limit ordinal logic mathematicians mathematics maximal element non-empty set notation notion number systems order isomorphism order relation ordered by magnitude ordered pairs ordered set Peano's axioms prime ideal Proof Let proper class properties Prove rational numbers reader real numbers result Section set theory successor set suppose surjection totally ordered transfinite sequence uncountable union verify VNB9 well-formed formula well-ordered sets well-ordering theorem Zorn's lemma
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Bibliographic information
| Title | Numbers, Sets and Axioms: The Apparatus of Mathematics |
| Author | A. G. Hamilton |
| Edition | illustrated, reprint |
| Publisher | Cambridge University Press, 1982 |
| ISBN | 0521287618, 9780521287616 |
| Length | 255 pages |
| Subjects | › Mathematics / Combinatorics Mathematics / General Mathematics / Logic Mathematics / Set Theory |
| Export Citation | BiBTeX EndNote RefMan |