Constructive reverse mathematics: an introduction

Transcription

1 Constructive reverse mathematics: an introduction Hajime Ishihara School of Information Science Japan Advanced Institute of Science and Technology (JAIST) Nomi, Ishikawa , Japan CMFP 2013, Nis, June, 2013

2 Contents A history of constructivism Minimal, intuitionistic and classical logics Axioms of intuitionism Axioms of constructive recursive mathematics Axioms of constructive mathematics Classical theorems Some mathematical consequences of axioms Classical reverse mathematics Constructive reverse mathematics (CRM) Logical principles CRM with countable choice CRM without countable choice

3 A history of constructivism History Logic Arithmetization of mathematics (Kronecker, 1987) 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)

4 Mathematical theory A mathematical theory consists of axioms describing mathematical objects in the theory logic being used to derive theorems from the axioms Two ways of generalizing mathematical theory: axioms commutative group group monoid logic classical logic intuitionistic logic minimal logic

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 Deduction 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.

7 Deduction (Basis) For each formula A, is a deduction with conclusion A and assumptions {A}. A

8 Deduction (Induction step) 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

9 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 E is a deduction with conclusion B and assumptions Γ 1 Γ 2.

10 Deduction (Induction step) If Γ 1 D 1 A B Γ 2 D 2 C Γ 3 D 3 C are deductions, then Γ 1 D 1 A B Γ 2 D 2 C C Γ 3 D 3 C E is a deduction with conclusion C and assumptions Γ 1 (Γ 2 \ {A}) (Γ 3 \ {B}). We write D 1 A B [A] D 2 C C [B] D 3 C E

11 Minimal logic D 2 A B A B I D 1 D A A B I r [A] D B A B I D B A B I l D A B A D 1 A B E r D 1 A B B [A] D 2 C C D A B B D 2 A [B] D 3 C E l E E

12 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).

13 Minimal logic We denote by Γ m A that there is a deduction in minimal logic with the conclusion A and the assumptions in Γ.

14 Example (minimal logic) [A B] [A] [ B] B E [ A] A I E [ (A B)] (A B) I E B I A B I (A B) ( A B) I E

15 Intuitionistc logic Intuitionistic logic is obtained from minimal logic by adding the intuitionistic absurdity rule: D A i We denote by Γ i A that there is a deduction in intuitionistic logic with the conclusion A and the assumptions in Γ. Note that Γ m A Γ i A.

16 Example (intuitionistic logic) [ A B] B [ (A B)] [ A] [A] B i A B A [ (A B)] B (A B) ( A B) (A B) [B] A B

17 Classical logic Classical logic is obtained from intuitionistic logic by strengthening the absurdity rule to the classical absurdity rule: [ A] D A c We denote by Γ c A that there is a deduction in classical logic with the conclusion A and the assumptions in Γ. Note that Γ i A Γ c A.

18 Example (classical logic) The double negation elimination (DNE): [ A] [ A] E A c A A I

19 Example (classical logic) The principle of excluded middle (PEM): [A] [ (A A)] A A I r E A I [ (A A)] A A I l E A A c

20 Example (classical logic) De Morgan s law (DML): [A] [B] [ (A B)] A B I E A I [ ( A B)] A B I r E B I [ ( A B)] A B I l E A B c (A B) A B I

21 Logical equivalences as equivalence relations Each logic defines an equivalence relation on the set of formulas by A B A B where {m, i, c}. We have and A [ A] m A m B A i B A c B A B [ (A B)] m A B [ (A B)] i (A B) [ A B] i (A B) [ A B] c

22 Notations m, n, i, j, k,... N a, b, c,... N a = the length of a a b = the concatenation of a and b a b c(a c = b) α, β, γ, δ,... N N α(n) = (α(0),..., α(n 1)) α a α( a ) = a {0, 1} = {a n < a (a(n) = 0 a(n) = 1)} {0, 1} N = {α n(α(n) = 0 α(n) = 1)}

23 Axioms of intuitionism The weak continuity for numbers (WC-N): α na(α, n) α mn β α(m)a(β, n) The fan theorem (FAN): α {0, 1} N na(α(n)) n α {0, 1} N k na(α(k)) We denote FAN for quantifier free A by FAN D.

24 Axioms of intuitionism (continuity principle) Let F be a function from N N into N. Then to compute F (α) (in finite step), it suffices to know finite information α(m) for some m. Therefore F must be continuous: α m β α(m)(f (α) = F (β)). Combining this with an axiom of choice we have WC-N: α na(α, n) F N NN αa(α, F (α)) α na(α, n) α mn β α(m)a(β, n).

25 Axioms of intuitionism (fan theorem) Suppose that A is quantifier free and upward closed, i.e. A(a) a b A(b). Let a T A(a). Then T is a tree, i.e. b T a b a T.

26 Axioms of intuitionism (fan theorem) The weak König lemma (WKL): Every infinite binary tree has an infinite path is expressed by n a {0, 1} ( a = n a T ) α {0, 1} N n(α(n) T ). It has a classical contraposition: α {0, 1} N na(α(n)) n a {0, 1} ( a = n A(a)) which is equivalent to FAN D : α {0, 1} N na(α(n)) n α {0, 1} N k na(α(k)).

27 Axioms of constructive recursive mathematics Extended Church s thesis (ECT 0 ): n[a(n) mb(n, m)] k n[a(n) m(t (k, n, m) B(n, U(m)))] where A is almost negative. Markov s principle (MP, Σ 0 1 -DNE): n(α(n) 0) n(α(n) 0)

28 Axioms of constructive recursive mathematics The principle (Church s thesis): each sequence of natural numbers is computable is expressed by an axiom α k n m(t (k, n, m) U(m) = α(n)) where T and U are Kleene s T -predicate and the result-extracting function, respectively.

29 Axioms of constructive recursive mathematics Combining this with an axiom of choice: n mb(n, m) α nb(n, α(n)) we have the arithmetical form of Church s thesis: n mb(n, m) k n m[t (k, n, m) B(n, U(m))] which has an extended form ECT 0 : n[a(n) mb(n, m)] k n[a(n) m(t (k, n, m) B(n, U(m)))] where A is almost negative.

30 Axioms of constructive mathematics 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))] The axiom of unique choice (AC!): x X!y YA(x, y) f Y X x XA(x, f (x)).

31 Relationship between axioms Proposition ECT 0 + FAN D and ECT 0 + WC-N. Proposition WKL FAN D and FAN D c WKL. Proposition ECT 0 + AC 1,0, where AC 1,0 is an axiom of choice: α na(α, n) F N NN αa(α, F (α))

32 Classical theorems Classical theorems on compactness: MCT: Every bounded monotone sequence of real numbers converges. SCT: Every sequence of a compact metric space has a convergent subsequence. HBT: Every (countable) open cover of a compact metric space has a finite subcover. CIT: Every family (sequence) of closed subsets of a compact metric space with finite intersection property has an intersection. MIN: Every uniformly continuous real function on a compact metric space attains its minimum.

33 Classical theorems Classical theorems on continuity: DCT: There exists a discontinuous mapping from N N into N. UCT: Every continuous mapping from a compact metric space into a (separable) metric space is uniformly continuous.

34 Some mathematical consequences of axioms Theorem (FAN) HBT. Theorem (FAN) If f : [0, 1] R is a uniformly continuous function such that f (x) > 0 for each x [0, 1], then inf(f ) > 0.

35 Some mathematical consequences of axioms Theorem (ECT 0 ) ST: There exists a bounded monotone sequence (q n ) n of rational numbers such that for each x R mk n m(2 k < x q n ). Theorem (ECT 0 ) There exists a uniformly continuous function f : [0, 1] R such that f (x) > 0 for each x [0, 1] and inf(f ) = 0.

36 Some mathematical consequences of axioms Theorem (WC-N or ECT 0 + MP) KLST: Every mapping from a complete separable metric space into a metric space is continuous. Theorem (FAN) UCT. Theorem (ECT 0 ) There exists a continuous mapping from {0, 1} N into N which is not uniformly continuous.

37 Classical reverse mathematics The Friedman-Simpson-program (classical reverse mathematics) is a formal mathematics using classical logic, assuming a very weak set existence axiom. main question is Which set existence axioms are needed to prove the theorems of ordinary mathematics?. many theorems have been classified by set existence axioms of various strengths.

38 Classical reverse mathematics The arithmetical comprehension axiom (ACA): X n(n X A) where A is an arithmetical formula. Theorem The following are equivalent. ACA MCT SCT

39 Classical reverse mathematics Theorem The following are equivalent. WKL HBT CIT MIN UCT

40 Constructive reverse mathematics (CRM) Since classical reverse mathematics is formalized with classical logic, we cannot classify theorems in intuitionism nor in constructive recursive mathematics which are inconsistent with classical mathematics distinguish theorems from their contrapositions.

41 Constructive reverse mathematics (CRM) Constructive mathematics is an informal mathematics using intuitionistic logic assuming some function existence axioms a core of the varieties of mathematics which can be extended to intuitionism (by adding WC-N and FAN) constructive recursive mathematics (by adding ECT 0 and MP) classical mathematics (by adding PEM).

42 Constructive reverse mathematics (CRM) The purpose of constructive reverse mathematics is to classify various theorems in intuitionistic, constructive recursive and classical mathematics by logical principles, function existence axioms and their combinations.

43 Logical principles The limited principle of omniscience (LPO, Σ 0 1 -PEM): n(α(n) 0) n(α(n) 0) The weak limited principle of omniscience (WLPO, Π 0 1 -PEM): n(α(n) 0) n(α(n) 0)

44 Logical principles The lesser limited principle of omnicience (LLPO, Σ 0 1 -DML): ( n(α(n) 0) n(β(n) 0)) n(α(n) 0) n(β(n) 0) Markov s principle for disjunction (MP, Π 0 1 -DML): ( n(α(n) 0) n(β(n) 0)) n(α(n) 0) n(β(n) 0)

45 Logical principles Proposition LPO WLPO LLPO MP. Proposition LPO MP MP. Proposition LPO WLPO + MP. Proposition ECT 0 + LLPO and WC-N + LLPO.

46 Logical principles Weak Markov s principle (WMP): β[ n(β(n) 0) n(β(n) α(n))] n(α(n) 0) Proposition MP WMP + MP. Proposition ECT 0 WMP and WC-N WMP.

47 CRM with countable choice (compactness) Definition A metric space is compact if it is complete and totally bounded. Theorem The following are equivalent. LPO MCT SCT

48 CRM with countable choice (compactness) Theorem The following are equivalent. LLPO WKL CIT MIN

49 CRM with countable choice (compactness) Definition A real function f on a metric space has at most one minimum if 0 < d(x, y) implies f (z) < f (x) f (z) < f (y) for some z. A binary tree T has at most one path if α β implies α(n) T β(n) T for some n. Theorem The following are equivalent. FAN D Every infinite binary tree with at most one path has an infinite path. Every uniformly continuous real function on a compact metric space with at most one minimum attains its minimum.

50 CRM with countable choice (continuity) Definition A mapping f between metric spaces is discontinuous if there exist δ > 0 and a sequence (x n ) n converging to a limit x such that d(f (x n ), f (x)) δ for all n. Theorem The following are equivalent. WLPO DCT

51 CRM with countable choice (continuity) Definition A mapping f between metric spaces is nondiscontinuous if x n x and d(f (x n ), f (x)) δ imply δ 0. Theorem The following are equivalent. WLPO Every mapping from a complete metric space into a metric space is nondiscontinuous.

52 CRM with countable choice (continuity) Definition A mapping f between metric spaces is strongly extensional if 0 < d(f (x), f (y)) implies 0 < d(x, y). Theorem The following are equivalent. MP Every mapping between metric space is strongly extensional.

53 CRM with countable choice (continuity) Definition A mapping f between metric spaces is sequentially continuous if x n x implies f (x n ) f (x). Theorem The following are equivalent. WMP Every mapping from a complete metric space into a metric space is strongly extensional. Every nondiscontinuous mapping from a complete metric space into a metric space is sequentially continuous.

54 CRM with countable choice (continuity) Definition A subset S of N is pseudobounded if lim n s n /n = 0 for each sequence (s n ) n in S. Theorem The following are equivalent. BD-N: Every countable pseudobounded subset is bounded. Every sequentially continuous mapping from a separable metric space into a metric space is continuous. Proposition ECT 0 + MP BD-N and WC-N BD-N.

55 CRM with countable choice (continuity) Theorem The following are equivalent. WLPO + WMP + BD-N KLST

56 CRM without countable choice The axiom of countable number choice (AC 00 ): m na(m, n) α ma(m, α(m)) Theorem The following are equivalent. LPO + Π 0 1 -AC 00 MCT SCT

57 CRM without countable choice Definition A sequence (F n ) n of closed subsets of {0, 1} N is given by σ : {0, 1} N N such that α F n k(σ(α(k), n) = 0. A sequence (F n ) n has finite intersection property if for each n α(α n F i ). i=0

58 CRM without countable choice The axiom of countable choice for disjunction (AC 0 ): m(a(m) B(m)) α {0, 1} N m[(α(m) = 0 A(m)) (α(m) = 1 B(m))] Theorem The followng are equivalent. LLPO + Π 0 1 -AC 0 WKL CIT: Every sequence of closed subsets of {0, 1} N with finite intersection property has an intersection.

59 CRM without countable choice Definition A sequence (G n ) n of open subsets of {0, 1} N is given by σ : {0, 1} N N such that A sequence (G n ) n is a cover if α G n k(σ(α(k), n) 0. α {0, 1} N (α n G n ).

60 CRM without countable choice Definition A mapping F : {0, 1} N N is representable if there exists γ : {0, 1} N such that α {0, 1} N n[ k < n(γ(α(k)) = 0) γ(α(n)) = F (α) + 1]. Theorem The followng are equivalent. FAN D HBT: Every open cover has a finite subcover. UCT: Every representable mapping is uniformly continuous.