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Formalizing Mathematics through Weyl's Predicative Foundations in Type Theory

This talk presents a formalization of mathematics leveraging different logical foundations within a type-theoretic framework, inspired by Hermann Weyl's predicative mathematics. We explore the implications of predicativity and the distinctions between impredicative and predicative concepts in set theory. Key examples include logic-enriched type theory and applications in proof assistants like Coq, focusing on the Fundamental Theorem of Algebra and the Four-Color Theorem. The discussion extends to the adequacy and uniformity of these foundational frameworks, aiming to showcase the richness of mathematical pluralism.

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Formalizing Mathematics through Weyl's Predicative Foundations 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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