Constructive (functional) analysis
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1 Constructive (functional) analysis Hajime Ishihara School of Information Science Japan Advanced Institute of Science and Technology (JAIST) Nomi, Ishikawa , Japan Proof and Computation, Fischbachau, 3 8 October, 2016
2 Contents (lectures 1 3) Intuitionistic logic Real numbers Metric spaces Normed and Banach spaces Hilbert spaces
3 Contents (lecture 1) The BHK interpretation Natural deduction Minimal logic Intuitionistic logic Classical logic Omniscience principles Number systems Real numbers Ordering relation Apartness and equality Arithmetical operations
4 A history of constructivism History Logic Arithmetization of mathematics (Kronecker, 1887) Three kinds of intuition (Poincaré, 1905) French semi-intuitionism (Borel, 1914) Intuitionism (Brouwer, 1914) Predicativity (Weyl, 1918) Finitism (Skolem, 1923; Hilbert-Bernays, 1934) Constructive recursive mathematics (Markov, 1954) Constructive mathematics (Bishop, 1967) Intuitionistic logic (Heyting, 1934; Kolmogorov, 1932)
5 Language We use the standard language of (many-sorted) first-order predicate logic based on primitive logical operators,,,,,. We introduce the abbreviations A A ; A B (A B) (B A).
6 The BHK interpretation The Brouwer-Heyting-Kolmogorov (BHK) interpretation of the logical operators is the following. A proof of A B is given by presenting a proof of A and a proof of B. A proof of A B is given by presenting either a proof of A or a proof of B. A proof of A B is a construction which transform any proof of A into a proof of B. Absurdity has no proof. A proof of xa(x) is a construction which transforms any t into a proof of A(t). A proof of xa(x) is given by presenting a t and a proof of A(t).
7 Natural Deduction System We shall use D, possibly with a subscript, for arbitrary deduction. We write Γ D A to indicate that D is deduction with conclusion A and assumptions Γ.
8 Deduction (Basis) For each formula A, is a deduction with conclusion A and assumptions {A}. A
9 Deduction (Induction step, I) If Γ D B is a deduction, then Γ D B A B I is a deduction with conclusion A B and assumptions Γ \ {A}. We write [A] D B A B I
10 Deduction (Induction step, ) If are deductions, then Γ 1 D 1 A B Γ 1 D 1 A B B Γ 2 D 2 A Γ 2 D 2 A is a deduction with conclusion B and assumptions Γ 1 Γ 2.
11 Example [A B] [A] [ B] B [ (A B)] (A B) I [ A] A I B I A B I (A B) ( A B) I
12 Minimal logic [A] D B A B I D 2 A B A B I D 1 D A A B I r D B A B I l D A B A D 1 A B B D 1 A B E r [A] D 2 C C D 2 A D A B B E l [B] D 3 C E
13 Minimal logic D A ya[x/y] I D A[x/t] xa I D xa A[x/t] E D 1 ya[x/y] C [A] D 2 C E In E and I, t must be free for x in A. In I, D must not contain assumptions containing x free, and y x or y FV(A). In E, D 2 must not contain assumptions containing x free except A, x FV(C), and y x or y FV(A).
14 Example [(A B) (A C)] [(A B) (A C)] E A B r E [A] A C l B C I B C A B C I (A B) (A C) (A B C) I [A]
15 Example [A B] [(A C) (B C)] A C C E r [A] C A B C I (A C) (B C) (A B C) I [(A C) (B C)] B C C E l E [B]
16 Example where x FV(A). [A xb] [A] xb B E A B I x(a B) I (A xb) x(a B) I
17 Example where x FV(A). [A B] [A] B [ x(a B)] xb I E xb A xb I x(a B) (A xb) I
18 Intuitionistic logic Intuitionistic logic is obtained from minimal logic by adding the intuitionistic absurdity rule (ex falso quodlibet). If Γ D is a deduction, then Γ D A i is a deduction with conclusion A and assumptions Γ.
19 Example [ A] [A] B i [ (A B)] A B I [B] [ A B] A I [ (A B)] A B I B B I (A B) I ( A B) (A B) I
20 Example [ A] [A] [A B] B i [B] B A B I A B ( A B) I E
21 Classical logic Classical logic is obtained from intuitionistic logic by strengthening the absurdity rule to the classical absurdity rule (reductio ad absurdum). If Γ D is a deduction, then Γ D A c is a deduction with conclusion A and assumption Γ \ { A}.
22 Example (classical logic) The double negation elimination (DNE): [ A] [ A] A c A A I
23 Example (classical logic) The principle of excluded middle (PEM): [A] [ (A A)] A A I r A I [ (A A)] A A I l A A c
24 Example (classical logic) De Morgan s law (DML): [A] [B] [ (A B)] A B I A I [ ( A B)] A B I r B I [ ( A B)] A B I l A B c (A B) A B I
25 RAA vs I c : deriving A by deducing absurdity ( ) from A. [ A] D A c I: deriving A by deducing absurdity ( ) from A. [A] D A I
26 Notations m, n, i, j, k,... N α, β, γ, δ,... N N 0 = λn.0 α # β n(α(n) β(n))
27 Omniscience principles The limited principle of omniscience (LPO, Σ 0 1 -PEM): α[α # 0 α # 0] The weak limited principle of omniscience (WLPO, Π 0 1 -PEM): α[ α # 0 α # 0] The lesser limited principle of omniscience (LLPO, Σ 0 1 -DML): αβ[ (α # 0 β # 0) α # 0 β # 0]
28 Markov s principle Markov s principle (MP, Σ 0 1 -DNE): α[ α # 0 α # 0] Markov s principle for disjunction (MP, Π 0 1 -DML): αβ[ ( α # 0 β # 0) α # 0 β # 0] Weak Markov s principle (WMP): α[ β( β # 0 β # α) α # 0]
29 Remark We may assume without loss of generality that α (and β) are ranging over binary sequences, nondecreasing sequences, sequences with at most one nonzero term, or sequences with α(0) = 0.
30 Relationship among principles MP zu uuuuuuuu J LPO J JJJJJJJJ% WLPO J JJJJJJJJJJJJJJJJJJJJJ LLPO $ WMP MP LPO WLPO + MP MP WMP + MP
31 Remark MP (and hencce WMP and MP ) holds in constructive recuresive mathematics. WMP holds in intuitionism.
32 CZF and choice axioms The materials in the lectures could be formalized in the constructive Zermelo-Fraenkel set theory (CZF) without the powerset axiom and the full separation axiom, together with the following choice axioms. The axiom of countable choice (AC 0 ): n y YA(n, y) f Y N na(n, f (n)) The axiom of dependent choice (DC): x X y XA(x, y) x X f X N [f (0) = x na(f (n), f (n + 1))]
33 Number systems The set Z of integers is the set N N with the equality (n, m) = Z (n, m ) n + m = n + m. The arithmetical relations and operations are defined on Z in a straightforwad way; natural numbers are embedded into Z by the mapping n (n, 0). The set Q of rationals is the set Z N with the equality (a, m) = Q (b, n) a (n + 1) = Z b (m + 1). The arithmetical relations and operations are defined on Q in a straightforwad way; integers are embedded into Q by the mapping a (a, 0).
34 Real numbers Definition A real number is a sequence (p n ) n of rationals such that mn ( p m p n < 2 m + 2 n). We shall write R for the set of real numbers as usual. Remark Rationals are embedded into R by the mapping p p = λn.p.
35 Ordering relation Definition Let < be the ordering relation between real numbers x = (p n ) n and y = (q n ) n defined by x < y n ( 2 n+2 < q n p n ). Proposition Let x, y, z R. Then (x < y y < x), x < y x < z z < y.
36 Ordering relation Proof. Let x = (p n ) n, y = (q n ) n and z = (r n ) n, and suppose that x < y. Then there exists n such that 2 n+2 < q n p n. Setting N = n + 3, either (p n + q n )/2 < r N or r N (p n + q n )/2. In the former case, we have 2 N+2 < 2 n+1 (2 (n+3) + 2 n ) < q n p n 2 = p n + q n p N < r N p N, 2 (p N p n ) and hence x < z. In the latter case, we have 2 N+2 < (2 (n+3) + 2 n ) + 2 n+1 < (q N q n ) + q n p n 2 = q N p n + q n q N r N, 2 and hence z < y.
37 Apartness and equality Definition We define the apartness #, the equality =, and the ordering relation between real numbers x and y by x # y (x < y y < x), x = y (x # y), x y (y < x). Lemma Let x, y, z R. Then x # y y # x, x # y x # z z # y.
38 Apartness and equality Proposition Let x, y, z R. Then x = x, x = y y = x, x = y y = z x = z. Proposition Let x, x, y, y R. Then x = x y = y x < y x < y, (x < y x = y y < x), x < y y < z x < z.
39 Apartness and equality Corollary Let x, x, y, y, z R. Then x = x y = y x # y x # y, x = x y = y x y x y, x y (x < y x = y), (x y y x), x y y x x = y, x < y y z x < z, x y y < z x < z, x y y z x z.
40 Apartness and equality Proposition xy R(x # y x = y) LPO, Proof. ( ): Let x = (p n ) n and y = (q n ) n, and define a binary sequence α by α(n) = 1 2 n+2 < q n p n. Then α # 0 x # y, and hence x # y x = y, by LPO. ( ): Let α be a binary sequence α with at most one nonzero term, and define a sequence (p n ) n of rationals by p n = n α(k) 2 k. k=0 Then x = (p n ) n R, and x # 0 α # 0. Therefore α # 0 α # 0, by x # 0 x = 0.
41 Apartness and equality Proposition xy R( x = y x = y) WLPO, xy R(x y y x) LLPO, xy R( x = y x # y) MP, xyz R( x = y x = z z = y) MP, xy R( z R( x = z z = y) x # y) WMP.
42 Arithmetical operations The arithmetical operations are defined on R in a straightforwad way. For x = (p n ), y = (q n ) R, define x + y = (p n+1 + q n+1 ); x = ( p n ); x = ( p n ); max{x, y} = (max{p n, q n });.
43 References Peter Aczel and Michael Rathjen, CST Book draft, 2010, rathjen/book.pdf. Errett Bishop, Foundations of Constructive Analysis, McGraw-Hill, New York, Errett Bishop and Douglas Bridges, Constructive Analysis, Springer-Verlag, Berlin, Douglas Bridges and Fred Richman, Varieties of Constructive Mathematics, Cambridge Univ. Press, London, Douglas Bridges and Luminiţa Vîţă, Techniques of Constructive Analysis, Springer, New York, D. van Dalen, Logic and Structure, 5th ed., Springer, London, A.S. Troelstra and D. van Dalen, Constructivism in Mathematics, An Introduction, Vol. I, North-Holland, Amsterdam, 1988.
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