Lectures on Constructive Mathematics. - Mathematics, Algorithms ...

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Metric spaces The elementary constructive notions associated with a metric space (X; ) are more or less as in classical mathematics. Note that when there are alternative classical de…nitions, these may not be equivalent constructively. In that case, we choose the most computationally informative notion. For example, we do not de…ne closed sets as complements of open sets: a closed set S is de…ned to be one that equals it closure S; in other words, S is closed in X if and only if all limits of sequences in S belong to S. We denote the open and closed balls in X with centre a and radius r > 0 by B(a; r) and B(a; r) respectively.
As in the classical theory, both X and ? are open, unions of open sets are open, and …nite intersections of open sets are open; in other words, the open sets form a topology on X. Likewise, both X and ? are closed, and arbitrary intersections of closed sets are closed. We cannot prove that the union of two closed sets is closed: in the metric space R, the intervals [ 1; 0] and [0; 1] are closed and their union is dense in [ 1; 1]; but if that union is closed, then a proposition equivalent to LLPO. 8 x2R (x 6 0 _ x > 0) ;
Page 1 and 2: 1958-1 Summer School and Conference
Page 3 and 4: Lecture 1
Page 5 and 6: Constructive proof of the existence
Page 7 and 8: Although the decimal expansion of h
Page 9 and 10: Why is this proof nonconstructive?
Page 11 and 12: If we want to prove something const
Page 13 and 14: The constructive vs nonconstructive
Page 15 and 16: Bishop-style constructive mathemati
Page 17 and 18: We do not restrict to a class of
Page 19 and 20: The BHK interpretation Modern intui
Page 21 and 22: I 9xP (x) : we have (i) an algorith
Page 23 and 24: The Goldbach conjecture (GC, 1742):
Page 25 and 26: The use of GC here is purely illust
Page 27 and 28: Suppose we have a constructive proo
Page 29 and 30: Here is another essentially noncons
Page 31 and 32: Consider the classically trivial pr
Page 33 and 34: The following elementary classical
Page 35 and 36: Note: classically valid statements
Page 37 and 38: We use these models of BISH to obta
Page 39 and 40: Note, though, that the statement If
Page 41 and 42: The last of these properties ensure
Page 43 and 44: Let x; y be real numbers. We say th
Page 45 and 46: Let S be a subset of R. An upper bo
Page 47: Lecture 2
Page 51 and 52: The complement of a set S in X is t
Page 53 and 54: Total boundedness is very important
Page 55 and 56: Another extremely important, though
Page 57 and 58: Let X be a totally bounded metric s
Page 59 and 60: A complete, totally bounded metric
Page 61 and 62: Proposition: Let Y be an inhabited
Page 63 and 64: A seminorm on a linear space X is m
Page 65 and 66: A mapping u between vector spaces X
Page 67 and 68: Proposition: A nonzero linear funct
Page 69 and 70: Inducting on the dimension, we can
Page 71 and 72: A subspace Y of a metric space X is
Page 73 and 74: The next lemma is crucial for the p
Page 75 and 76: Two subsets S; T of an inner produc
Page 77 and 78: Classically, using (an equivalent o
Page 79 and 80: The Riesz representation theorem fo
Page 81 and 82: An operator on a Hilbert space H is
Page 83 and 84: If T is jointed, then T is an opera
Page 85 and 86: A linear mapping T between normed s

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