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Weyl’s predicative math in type theory Zhaohui Luo Dept of Computer Science Royal Holloway, Univ of London (Joint work with Robin Adams)

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April 20062 Formalisation of mathematics with different logical foundations in a type-theoretic framework

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April 20063 This talk Maths based on different logical foundations Weyl’s predicative mathematics Type-theoretic framework Example: logic-enriched TT with classical logic Predicativity Impredicative and predicative notions of set Formalisations Real number system, predicatively and impredicatively

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April 20064 I. Applications of TT to formalisation of maths Formalisation in TT-based proof assistants Examples in Coq: Fundamental Theorem of Algebra Four-colour Theorem Maths with different logical foundations Variety of maths, all legacies (mathematical “pluralism”) Adequacy in formalisation? Uniform framework? Type theory and associated technology Not just for constructive math Also for classical math and other maths

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April 20065 Maths with different logical foundations: examples Consider the “combinations” of the following and their “negations”: (C)Classical logic (I) Impredicative definitions We would have (CI) Ordinary (classical, impredicative) math Classical set theory/simple type theory, HOL/Isabelle (C°I°)Predicative constructive math Martin-Löf’s TT, ALF/Agda/NuPRL (C°I)Impredicative constructive math Constructions/CID/ECC/UTT, Coq/Lego/Plastic (CI°)Predicative classical math Weyl, Feferman, Simpson, … Uniform foundational framework for formalisation?

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April 20066 Weyl’s predicative mathematics H. Weyl. The Continuum. (Das Kontinuum.) 1918. Historical development (paradox etc.) The notion of category Predicative development of the real number system Weyl/Feferman/Simpson’s work on predicativity Predicativity E.g., { x | φ(x) } with φ being “arithmetical” (without quantification over sets) Feferman’s development on “predicativism” Simpson’s work on reverse mathematics

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April 20067 II. Logic-enriched type theories in LFs Logic-enriched type theory Aczel & Gambino (LTT in the intuitionistic setting) [AG02,06] c.f. separation of logical propositions and data types in ECC/UTT [Luo90,94] Type-theoretic framework for mathematical “pluralism” Logic-enriched TTs in a logical framework: Logic Types \ / Logical Framework

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April 20068 An example: T T = LF + Classical FOL + Ind types/universes Classical Ind types FOL + universes \/ LF

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April 20069 Classical FOL (specified in a logical framework) Propositions (note: LF should be “extended” with Prop and Prf) Prop kind Prf(P) kind [P : Prop] Logical operators P Q : Prop [P : Prop, Q : Prop] [A,P] : Prop [A : Type, P[x:A] : Prop] ¬P : Prop [P : Prop] DN[P,p] : Prf(P) [P:Prop, p:Prf(¬¬P)]

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April 200610 Types Inductive types/families e.g. Nats, Trees, … (as in TTs such as UTT) Induction Rule: elimination over propositions. Example: the natural numbers N : Type, 0 : N, succ[n] : N [n : N] Elimination over types: Elim T [C,c,f,n] : C[n], for C[n] : Type [n : N] Plus computational rules for Elim T : eg, Elim T [C,c,f,succ(n)] = f[n,Elim T [C,c,f,n]] Induction over propositions: Elim P [P,c,f,n] : P[n], for P[n] : Prop [n : N]

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April 200611 Relative consistency Theorem (relative consistency of T ) T is logically consistent w.r.t. ZF.

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April 200612 III. Formalisation Consider Classical logic T \ / LF with T = Inductive types + Impredicative sets (I) Predicative sets (I°)

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April 200613 Impredicative notion of set Typed sets, impredicatively: Set[A:Type] : Type set[A:Type,P[x:A]:Prop] : Set[A] in[A:Type,a:A,S:Set[A]] : Prop in[A,a,set[A,P]] = P[a] : Prop Every set has a “base type” (or “category”) Sets are given by characteristic propositional functions { x : A | P(x) } – set(A,P) s S – in(A,s,S) One can formulate powersets as …

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April 200614 Predicative notion of set Type universe and propositional universe type : Type and T[a:type] : Type (universe of “small types”) prop : Prop and V[p:prop] : Prop (universe of “small propositions”) [a:type,p[x:T[a]]:prop] : prop and V[ [a,p]] = [T[a],V◦p] : Prop Predicative notion of set Set[A:Type] : Type set[A:Type,p[x:A]:prop] : Set[A] in[A:Type,x:A,S:Set[A]] : prop in[A,x,set[A,p]] = p[x] : prop

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April 200615 Formalisation in Plastic Plastic (Callaghan [CL01]) Plastic: proof assistant, implementing a logical framework Extending Plastic with “Prop” etc. Formalisation Weyl’s predicative development Nats, Integers, Rationals, and Dedekind cuts. Completion and LUB theorems for real numbers. Other features Types as informal “categories” Typed sets Setoids Comparison between predicative and impredicative developments

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Mathematical Induction. F(1) = 1; F(n+1) = F(n) + (2n+1) for n≥1 100816449362516941 10987654321 F(n) n F(n) =n 2 for all n ≥ 1 Prove it!

Mathematical Induction. F(1) = 1; F(n+1) = F(n) + (2n+1) for n≥1 100816449362516941 10987654321 F(n) n F(n) =n 2 for all n ≥ 1 Prove it!

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