Constructive (functional) analysis Hajime Ishihara School of Information Science Japan Advanced Institute of Science and Technology (JAIST) Nomi, Ishikawa 923-1292, Japan Proof and Computation, Fischbachau, 3 { 8 October, 2016 Contents (lectures 1 { 3) I Intuitionistic logic I Real numbers I Metric spaces I Normed and Banach spaces I Hilbert spaces Contents (lecture 1) I The BHK interpretation I Natural deduction I Minimal logic I Intuitionistic logic I Classical logic I Omniscience principles I Number systems I Real numbers I Ordering relation I Apartness and equality I Arithmetical operations A history of constructivism I History I Arithmetization of mathematics (Kronecker, 1887) I Three kinds of intuition (Poincar´e,1905) I French semi-intuitionism (Borel, 1914) I Intuitionism (Brouwer, 1914) I Predicativity (Weyl, 1918) I Finitism (Skolem, 1923; Hilbert-Bernays, 1934) I Constructive recursive mathematics (Markov, 1954) I Constructive mathematics (Bishop, 1967) I Logic I Intuitionistic logic (Heyting, 1934; Kolmogorov, 1932) Language We use the standard language of (many-sorted) first-order predicate logic based on I primitive logical operators ^; _; !; ?; 8; 9. We introduce the abbreviations I :A ≡ A !?; I A $ B ≡ (A ! B) ^ (B ! A). The BHK interpretation The Brouwer-Heyting-Kolmogorov (BHK) interpretation of the logical operators is the following. I A proof of A ^ B is given by presenting a proof of A and a proof of B. I A proof of A _ B is given by presenting either a proof of A or a proof of B. I A proof of A ! B is a construction which transform any proof of A into a proof of B. I Absurdity ? has no proof. I A proof of 8xA(x) is a construction which transforms any t into a proof of A(t). I A proof of 9xA(x) is given by presenting a t and a proof of A(t). 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. Deduction (Basis) For each formula A, A is a deduction with conclusion A and assumptions fAg. Deduction (Induction step, !I) If Γ D B is a deduction, then Γ D B !I A ! B is a deduction with conclusion A ! B and assumptions Γ n fAg. We write [A] D B !I A ! B Deduction (Induction step, !E) If Γ1 Γ2 D1 D2 A ! B A are deductions, then Γ1 Γ2 D1 D2 A ! B A !E B is a deduction with conclusion B and assumptions Γ1 [ Γ2. Example [A ! B][A] !E [:B] B !E ? !I [::(A ! B)] :(A ! B) !E ? !I [::A] :A !E ? !I ::B !I ::A ! ::B !I ::(A ! B) ! (::A ! ::B) Minimal logic [A] D D1 D2 B A ! B A !I !E A ! B B D1 D2 D D A B A ^ B A ^ B ^I ^E ^E A ^ B A r B l [A] [B] D D D1 D2 D3 A B A _ B C C _I _I _E A _ B r A _ B l C Minimal logic D D A 8xA 8I 8E 8yA[x=y] A[x=t] [A] D D1 D2 A[x=t] 9yA[x=y] C 9I 9E 9xA C I In 8E and 9I, t must be free for x in A. I In 8I, D must not contain assumptions containing x free, and y ≡ x or y 62 FV(A). I In 9E, D2 must not contain assumptions containing x free except A, x 62 FV(C), and y ≡ x or y 62 FV(A). Example [(A ! B) ^ (A ! C)] [(A ! B) ^ (A ! C)] ^E ^E A ! B r [A] A ! C l [A] !E !E B C ^I B ^ C !I A ! B ^ C !I (A ! B) ^ (A ! C) ! (A ! B ^ C) Example [(A ! C) ^ (B ! C)] [(A ! C) ^ (B ! C)] ^E ^E A ! C r [A] B ! C l [B] !E !E [A _ B] C C _E C !I A _ B ! C !I (A ! C) ^ (B ! C) ! (A _ B ! C) Example [A ! 8xB][A] !E 8xB 8E B !I A ! B 8I 8x(A ! B) !I (A ! 8xB) ! 8x(A ! B) where x 62 FV(A). Example [A ! B][A] !E B 9I [9x(A ! B)] 9xB 9E 9xB !I A ! 9xB !I 9x(A ! B) ! (A ! 9xB) where x 62 FV(A). 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. Example [:A][A] !E ? ? B i !I [:(A ! B)] A ! B [B] !E !I ? [:(A ! B)] A ! B !I !E [::A ! ::B] ::A ? !E !I ::B :B ? !I ::(A ! B) !I (::A ! ::B) ! ::(A ! B) Example [:A][A] !E ? ? [A _ B] B i [B] _E B !I :A ! B !I A _ B ! (:A ! B) 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 n f:Ag. Example (classical logic) The double negation elimination (DNE): [::A][:A] !E ? ? A c !I ::A ! A Example (classical logic) The principle of excluded middle (PEM): [A] _I [:(A _:A)] A _:A r !E ? !I :A _I [:(A _:A)] A _:A l !E ? ? A _:A c Example (classical logic) De Morgan's law (DML): [A][B] ^I [:(A ^ B)] A ^ B !E ? !I :A _I [:(:A _:B)] :A _:B r !E ? !I :B _I [:(:A _:B)] :A _:B l !E ? ? :A _:B c !I :(A ^ B) !:A _:B 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 ? !I :A Notations I m; n; i; j; k;::: 2 N N I α;β;γ;δ;::: 2 N I 0 = λn:0 I α # β , 9n(α(n) 6= β(n)) Omniscience principles 0 I The limited principle of omniscience (LPO,Σ1-PEM): 8α[α # 0 _:α # 0] 0 I The weak limited principle of omniscience (WLPO,Π1-PEM): 8α[::α # 0 _:α # 0] 0 I The lesser limited principle of omniscience (LLPO,Σ1-DML): 8αβ[:(α # 0 ^ β # 0) !:α # 0 _:β # 0] Markov's principle 0 I Markov's principle (MP,Σ1-DNE): 8α[::α # 0 ! α # 0] _ 0 I Markov's principle for disjunction (MP ,Π1-DML): 8αβ[:(:α # 0 ^ :β # 0) ! ::α # 0 _ ::β # 0] I Weak Markov's principle (WMP): 8α[8β(::β # 0 _ ::β # α) ! α # 0] Remark We may assume without loss of generality that α (and β) are ranging over I binary sequences, I nondecreasing sequences, I sequences with at most one nonzero term, or I sequences with α(0) = 0. Relationship among principles uLPOJJ uu JJ uu JJ uu JJ uz u J% MP JJ WLPO JJ JJ JJ JJ JJ JJ JJ LLPO JJ JJ JJ J$ WMP MP_ I LPO , WLPO + MP _ I MP , WMP + MP Remark _ I MP (and hencce WMP and MP ) holds in constructive recuresive mathematics. I WMP holds in intuitionism. 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. I The axiom of countable choice (AC0): 8n9y 2 YA(n; y) ! 9f 2 Y N8nA(n; f (n)) I The axiom of dependent choice (DC): 8x 2 X 9y 2 XA(x; y)! 8x 2 X 9f 2 X N[f (0) = x ^ 8nA(f (n); f (n + 1))] Number systems I The set Z of integers is the set N × N with the equality 0 0 0 0 (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 7! (n; 0). I 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 7! (a; 0). Real numbers Definition A real number is a sequence (pn)n of rationals such that −m −n 8mn jpm − pnj < 2 + 2 : We shall write R for the set of real numbers as usual. Remark Rationals are embedded into R by the mapping p 7! p∗ = λn:p. Ordering relation Definition Let < be the ordering relation between real numbers x = (pn)n and y = (qn)n defined by −n+2 x < y , 9n 2 < qn − pn : Proposition Let x; y; z 2 R. Then I :(x < y ^ y < x), I x < y ! x < z _ z < y. Ordering relation Proof. Let x = (pn)n, y = (qn)n and z = (rn)n, and suppose that x < y. −n+2 Then there exists n such that 2 < qn − pn: Setting N = n + 3, either (pn + qn)=2 < rN or rN ≤ (pn + qn)=2. In the former case, we have q − p 2−N+2 < 2−n+1 − (2−(n+3) + 2−n) < n n − (p − p ) 2 N n p + q = n n − p < r − p ; 2 N N N and hence x < z. In the latter case, we have q − p 2−N+2 < −(2−(n+3) + 2−n) + 2−n+1 < (q − q ) + n n N n 2 p + q = q − n n ≤ q − r ; N 2 N N and hence z < y.