Weyl

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Weyls 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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• Formalisation of mathematics
• with different logical foundations
• in a type-theoretic framework

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This talk

• Maths based on different logical foundations
• Weyls 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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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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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
• (CI) Predicative constructive math
• Martin-Löfs TT, ALF/Agda/NuPRL
• (CI) 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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Weyls 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/Simpsons work on predicativity
• Predicativity
• E.g., x f(x) with f being arithmetical
  (without quantification over sets)
• Fefermans development on predicativism
• Simpsons work on reverse mathematics

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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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An example T

• T LF Classical FOL Ind types/universes
• Classical Ind types
• FOL universes
• \ /
• \ /
• LF

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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, PxA Prop
• P Prop P Prop
• DNP,p Prf(P) PProp, pPrf(P)

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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, succn N n N
• Elimination over types
• ElimTC,c,f,n Cn, for Cn Type n N
• Plus computational rules for ElimT eg,
• ElimTC,c,f,succ(n) fn,ElimTC,c,f,n
• Induction over propositions
• ElimPP,c,f,n Pn, for Pn Prop n N

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Relative consistency

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

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III. Formalisation

• Consider
• Classical logic T
• \ /
• \ /
• LF
• with
• T Inductive types Impredicative sets (I)
• Predicative sets (I)

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Impredicative notion of set

• Typed sets, impredicatively
• SetAType Type
• setAType,PxAProp SetA
• inAType,aA,SSetA Prop
• inA,a,setA,P Pa 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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Predicative notion of set

• Type universe and propositional universe
• type Type and Tatype Type (universe of
  small types)
• prop Prop and Vpprop Prop (universe of
  small propositions)
• ?atype,pxTaprop prop and V?a,p
  ?Ta,V?p Prop
• Predicative notion of set
• SetAType Type
• setAType,pxAprop SetA
• inAType,xA,SSetA prop
• inA,x,setA,p px prop

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Formalisation in Plastic

• Plastic (Callaghan CL01)
• Plastic proof assistant, implementing a logical
  framework
• Extending Plastic with Prop etc.
• Formalisation
• Weyls 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