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Weyl’s predicative math in type theory

Weyl’s predicative math in type theory. Zhaohui Luo Dept of Computer Science Royal Holloway, Univ of London (Joint work with Robin Adams). Formalisation of mathematics with different logical foundations in a type-theoretic framework. This talk.

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Weyl’s predicative math in type theory

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

  2. Formalisation of mathematics with different logical foundations in a type-theoretic framework

  3. 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

  4. 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

  5. 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?

  6. 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

  7. 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

  8. An example: T T = LF + Classical FOL + Ind types/universes Classical Ind types FOL + universes \ / \ / LF

  9. 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)]

  10. 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: • ElimT[C,c,f,n] : C[n], for C[n] : Type [n : N] • Plus computational rules for ElimT: eg, ElimT[C,c,f,succ(n)] = f[n,ElimT[C,c,f,n]] • Induction over propositions: • ElimP[P,c,f,n] : P[n], for P[n] : Prop [n : N]

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

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

  13. 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 …

  14. 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

  15. 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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