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
April 2006
2
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

April 2006
Real number system, predicatively and impredicatively
3
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


April 2006
Not just for constructive math
Also for classical math and other maths
4
Maths with different logical foundations: examples
Consider the “combinations” of the following and their “negations”:
(C)
(I)
Classical logic
Impredicative definitions
We would have

(CI)

(C°I°)

(C°I)

(CI°)
Ordinary (classical, impredicative) math
Classical set theory/simple type theory, HOL/Isabelle
Predicative constructive math
Martin-Löf’s TT, ALF/Agda/NuPRL
Impredicative constructive math
Constructions/CID/ECC/UTT, Coq/Lego/Plastic
Predicative classical math
Weyl, Feferman, Simpson, …
Uniform foundational framework for formalisation?
April 2006
5
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)


April 2006
Feferman’s development on “predicativism”
Simpson’s work on reverse mathematics
6
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
April 2006
7
An example: T
T = LF + Classical FOL + Ind types/universes
Classical
FOL
\
\
Ind types
+ universes
/
/
LF
April 2006
8
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)
April 2006
[P:Prop, p:Prf(¬¬P)]
9
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],
April 2006
for P[n] : Prop [n : N]
10
Relative consistency
Theorem (relative consistency of T)
T is logically consistent w.r.t. ZF.
April 2006
11
III. Formalisation
Consider
Classical logic T
\
/
\ /
LF
with
T = Inductive types +
April 2006
Impredicative sets
Predicative sets
(I)
(I°)
12
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)

April 2006
One can formulate powersets as …
13
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
April 2006
14
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




April 2006
Types as informal “categories”
Typed sets
Setoids
Comparison between predicative and impredicative developments
15