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

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Conclusion: The statement 8 x2R (x > 0 _ x 6 0) implies LLPO and is therefore essentially nonconstructive.
The following elementary classical statements also turn out to be nonconstructive. B Each real number x is either rational or irrational (that is, x 6= r for each rational number r). To see this, consider x = 1X n=1 1 an ; n! where a is any increasing binary sequence. This is equivalent to LPO. B Each real number x has a binary expansion. Note that the standard interval-halving argument for “constructing” binary expansions does not work, since we cannot necessarily decide, for a given number x between 0 and 1; whether x > 1=2 or x 6 1=2: In fact, the existence of binary expansions is equivalent to LLPO.
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: Consider the classically trivial pr
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 and 48: Lecture 2
Page 49 and 50: As in the classical theory, both X
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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Lectures on Constructive Mathematics. - Mathematics, Algorithms ...

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