Varieties of Constructive MathematicsThis is an introduction to, and survey of, the constructive approaches to pure mathematics. The authors emphasise the viewpoint of Errett Bishop's school, but intuitionism. Russian constructivism and recursive analysis are also treated, with comparisons between the various approaches included where appropriate. Constructive mathematics is now enjoying a revival, with interest from not only logicans but also category theorists, recursive function theorists and theoretical computer scientists. This account for non-specialists in these and other disciplines. |
Contents
Foundations of Constructive Mathematics | 1 |
Basic constructions | 6 |
Informal intuitionistic logic | 10 |
Choice axioms | 11 |
Real numbers | 12 |
Problems | 14 |
Notes | 16 |
2 Constructive Analysis | 18 |
3 Splitting fields | 79 |
4 Uniqueness of splitting fields | 82 |
5 Finitely presented modules | 87 |
6 Noetherian rings | 91 |
Problems | 97 |
Notes | 99 |
5 Intuitionism | 103 |
2 Continuous choice | 106 |
2 Baires theorem revisited | 21 |
3 Located subsets | 26 |
4 Totally Bounded Spaces | 28 |
5 Bounded Linear Maps | 34 |
6 Compactly Generated Banach Spaces | 41 |
Problems | 44 |
Notes | 47 |
1 Progranming Systems and Omniscience Principles | 49 |
2 Continuity and intermediate values | 54 |
3 Speckers Sequence | 58 |
4 The HeineBorel Theorem | 60 |
5 Moduli of continuity and cozero sets | 64 |
6 Ceitins theorem | 67 |
Problems | 71 |
Notes | 73 |
1 General considerations | 75 |
2 Factoring | 76 |
3 Uniform continuity | 110 |
4 The creating subject and Markovs principle | 116 |
Problems | 117 |
Notes | 119 |
1 The Three Varieties | 120 |
2 Positivevalued Continuous Functions | 122 |
Problems | 129 |
Notes | 130 |
1 Intuitionistic propositional calculus | 131 |
2 Predicate calculus | 134 |
3 The sheaf model CX | 138 |
4 Presheaf topos models | 141 |
Problems | 143 |
Notes | 144 |
| 146 | |
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Common terms and phrases
algorithm axiom of choice Baire's theorem Banach space binary sequence BISH bounded linear map Brouwer Brouwerian counterexample Brouwerian example Ceitin's theorem Chapter choose compact complete metric space Construct a Brouwerian constructive mathematics continuous function converges Corollary countable choice countable discrete field cozero sets creating subject defined detachable subset elements equivalent exists factor fan theorem finite set finite-dimensional finitely enumerable finitely generated submodules finitely presented follows function f Hilbert basis theorem induction interval intuitionistic logic irreducible isomorphic Kripke model left ideal Lemma Let linear functional located Markov's principle metric complement natural numbers Noetherian ring nonempty nonnegative nonvoid nonzero normable normed linear space partial function pointwise continuous polynomial positive integer problem propositional calculus prove quotient rational numbers real number regular Cauchy sequence RUSS S₁ splitting fields statement theory topos models totally bounded space truth value uniform uniformly continuous map whence λη