Constructive analysis Philosophy, Proof and Fundamentals Hajime Ishihara School of Information Science Japan Advanced Institute of Science and Technology (JAIST) Nomi, Ishikawa 923-1292, Japan Interval Analysis and Constructive Mathematics, Oaxaca, 13 { 18 November, 2016 Contents I The BHK interpretation I Natural deduction I Omniscience principles I Number systems I Real numbers I Ordering relation I Apartness and equality I Arithmetical operations I Cauchy competeness I Order completeness I The intermediate value theorem 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) Mathematical theory A mathematical theory consists of I axioms describing mathematical objects in the theory, such as I natural numbers, I sets, I groups, etc. I logic being used to derive theorems from the axioms objects logic Interval analysis intervals classical logic Constructive analysis arbitrary reals intuitionistic logic Computable analysis computable reals classical logic 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.