A categorical structure of realizers for the Minimalist Foundation

Transcription

1 A categorical structure of realizers for the Minimalist Foundation S.Maschio (joint work with M.E.Maietti) Department of Mathematics University of Padua TACL 2015 Ischia

2 The Minimalist Foundation Many foundations in (constructive) mathematics...

3 The Minimalist Foundation Many foundations in (constructive) mathematics...

4 The Minimalist Foundation Many foundations in (constructive) mathematics... Necessity of a common core:

5 The Minimalist Foundation Many foundations in (constructive) mathematics... Necessity of a common core: the minimalist foundation!

6 The Minimalist Foundation Many foundations in (constructive) mathematics... Necessity of a common core: the minimalist foundation! (Maietti, Sambin 2005)

7 The formal system of the Minimalist Foundation (Maietti (2009)):

8 The formal system of the Minimalist Foundation (Maietti (2009)): - 2-level theory based on versions of Martin Löf Type Theory;

9 The formal system of the Minimalist Foundation (Maietti (2009)): - 2-level theory based on versions of Martin Löf Type Theory; - an intensional level (mtt): computational content of proofs;

10 The formal system of the Minimalist Foundation (Maietti (2009)): - 2-level theory based on versions of Martin Löf Type Theory; - an intensional level (mtt): computational content of proofs; - an extensional level (emtt): where to develop ordinary mathematics.

11 The intensional level has four sorts of types:

12 The intensional level has four sorts of types: - sets: basic N 0, N 1 and constructors Π, Σ, + and List, all small propositions;

13 The intensional level has four sorts of types: - sets: basic N 0, N 1 and constructors Π, Σ, + and List, all small propositions; - collections: all sets and propositions, the collection of (codes for) small propositions prop s, A prop s with A set, constructor Σ;

14 The intensional level has four sorts of types: - sets: basic N 0, N 1 and constructors Π, Σ, + and List, all small propositions; - collections: all sets and propositions, the collection of (codes for) small propositions prop s, A prop s with A set, constructor Σ; - propositions: and closed under connectives,,, collection bounded quantifiers and Id in collections.

15 The intensional level has four sorts of types: - sets: basic N 0, N 1 and constructors Π, Σ, + and List, all small propositions; - collections: all sets and propositions, the collection of (codes for) small propositions prop s, A prop s with A set, constructor Σ; - propositions: and closed under connectives,,, collection bounded quantifiers and Id in collections. - small propositions are like propositions with only set bounded quantifiers and Id relative to sets, it contains decodings τ(p) for p prop s.

16 emtt= extensional version of mtt (includes extensionality of functions)

17 emtt= extensional version of mtt (includes extensionality of functions) + power-collections of sets

18 emtt= extensional version of mtt (includes extensionality of functions) + power-collections of sets + effective quotient sets

19 emtt= extensional version of mtt (includes extensionality of functions) + power-collections of sets + effective quotient sets + proof-irrelevance of propositions

20 emtt= extensional version of mtt (includes extensionality of functions) + power-collections of sets + effective quotient sets + proof-irrelevance of propositions it is interpreted in mtt via a quotient completion

21 Realizability Constructive mathematics =

22 Realizability Constructive mathematics = implicit computational mathematics =

23 Realizability Constructive mathematics = implicit computational mathematics = abstract mathematics with computational interpretation

24 Realizability Constructive mathematics = implicit computational mathematics = abstract mathematics with computational interpretation (realizability)

25 Kleene s realizability interpretation of connectives and quantifiers (for Heyting arithmetic)

26 Kleene s realizability interpretation of connectives and quantifiers (for Heyting arithmetic) r t = s is t = s r = 0

27 Kleene s realizability interpretation of connectives and quantifiers (for Heyting arithmetic) r t = s is t = s r = 0 r φ ψ is p 1 (r) φ p 2 (r) ψ

28 Kleene s realizability interpretation of connectives and quantifiers (for Heyting arithmetic) r t = s is t = s r = 0 r φ ψ is p 1 (r) φ p 2 (r) ψ r φ ψ is (p 1 (r) = 0 p 2 (r) φ) (p 1 (r) = 1 p 2 (r) ψ)

29 Kleene s realizability interpretation of connectives and quantifiers (for Heyting arithmetic) r t = s is t = s r = 0 r φ ψ is p 1 (r) φ p 2 (r) ψ r φ ψ is (p 1 (r) = 0 p 2 (r) φ) (p 1 (r) = 1 p 2 (r) ψ) r φ ψ is x(x φ {r}(x) ψ)

30 Kleene s realizability interpretation of connectives and quantifiers (for Heyting arithmetic) r t = s is t = s r = 0 r φ ψ is p 1 (r) φ p 2 (r) ψ r φ ψ is (p 1 (r) = 0 p 2 (r) φ) (p 1 (r) = 1 p 2 (r) ψ) r φ ψ is x(x φ {r}(x) ψ) r xφ(x) is p 2 (x) φ(p 1 (x))

31 Kleene s realizability interpretation of connectives and quantifiers (for Heyting arithmetic) r t = s is t = s r = 0 r φ ψ is p 1 (r) φ p 2 (r) ψ r φ ψ is (p 1 (r) = 0 p 2 (r) φ) (p 1 (r) = 1 p 2 (r) ψ) r φ ψ is x(x φ {r}(x) ψ) r xφ(x) is p 2 (x) φ(p 1 (x)) r xφ(x) is x({r}(x) φ(x))

32 Kleene s realizability interpretation of connectives and quantifiers (for Heyting arithmetic) r t = s is t = s r = 0 r φ ψ is p 1 (r) φ p 2 (r) ψ r φ ψ is (p 1 (r) = 0 p 2 (r) φ) (p 1 (r) = 1 p 2 (r) ψ) r φ ψ is x(x φ {r}(x) ψ) r xφ(x) is p 2 (x) φ(p 1 (x)) r xφ(x) is x({r}(x) φ(x)) mtt extends HA.

33 Kleene s realizability interpretation of connectives and quantifiers (for Heyting arithmetic) r t = s is t = s r = 0 r φ ψ is p 1 (r) φ p 2 (r) ψ r φ ψ is (p 1 (r) = 0 p 2 (r) φ) (p 1 (r) = 1 p 2 (r) ψ) r φ ψ is x(x φ {r}(x) ψ) r xφ(x) is p 2 (x) φ(p 1 (x)) r xφ(x) is x({r}(x) φ(x)) mtt extends HA. We model mtt by extending Kleene realizability interpretation.

34 ID 1 1 Theory of Inductive definitions:

35 ID 1 1 Theory of Inductive definitions: consists of Peano Arithmetic

36 ID 1 1 Theory of Inductive definitions: consists of Peano Arithmetic + fix points for positive paremeter-free operators

37 ID 1 1 Theory of Inductive definitions: consists of Peano Arithmetic + fix points for positive paremeter-free operators 2 Same proof theoretical strength as first-order Martin Löf type theory with one universe

38 ID 1 1 Theory of Inductive definitions: consists of Peano Arithmetic + fix points for positive paremeter-free operators 2 Same proof theoretical strength as first-order Martin Löf type theory with one universe

39 Collections of ID 1 A = ( A, A )

40 Collections of ID 1 A = ( A, A ) A = {x φ A (x)} is a (first-order) definable class of ID 1

41 Collections of ID 1 A = ( A, A ) A = {x φ A (x)} is a (first-order) definable class of ID 1 xεa is an abbreviation for φ A (x).

42 Collections of ID 1 A = ( A, A ) A = {x φ A (x)} is a (first-order) definable class of ID 1 xεa is an abbreviation for φ A (x). x A y is a (first-order) ID 1 -definable equivalence relation on A

43 Operations An arrow from A to B is an operation (a computable function):

44 Operations An arrow from A to B is an operation (a computable function): NOT a function (a single valued relation)!!!

45 Operations An arrow from A to B is an operation (a computable function): NOT a function (a single valued relation)!!! [n] A,B : A B

46 Operations An arrow from A to B is an operation (a computable function): NOT a function (a single valued relation)!!! [n] A,B : A B n is a numeral

47 Operations An arrow from A to B is an operation (a computable function): NOT a function (a single valued relation)!!! [n] A,B : A B n is a numeral x A y ID1 {n}(x) B {n}(y)

48 Operations An arrow from A to B is an operation (a computable function): NOT a function (a single valued relation)!!! [n] A,B : A B n is a numeral x A y ID1 {n}(x) B {n}(y) n A,B m if and only if xεa ID1 {n}(x) B {m}(x)

49 Proof irrelevance for all A, f, g A f g P f = g

50 Proof irrelevance for all A, f, g A f g P f = g equivalent to

51 Proof irrelevance for all A, f, g A f g P f = g equivalent to xεp yεp ID1 x P y

52 Proof irrelevance for all A, f, g A f g P f = g equivalent to xεp yεp ID1 x P y Propositions of ÎD 1 = def proof-irrelevant collections of ÎD 1

53 Proof irrelevance for all A, f, g A f g P f = g equivalent to xεp yεp ID1 x P y Propositions of ÎD 1 = def proof-irrelevant collections of ÎD 1 Propositions trivial quotients of the collections of their realizers

54 Universes An extended realizability interpretation of sets and small propositions of mtt can be encoded using fix points.

55 Universes An extended realizability interpretation of sets and small propositions of mtt can be encoded using fix points. collections of ÎD 1 of (codes for) sets and small propositions of mtt: US and USP.

56 Universes An extended realizability interpretation of sets and small propositions of mtt can be encoded using fix points. collections of ÎD 1 of (codes for) sets and small propositions of mtt: US and USP. f = [a] : 1 US(P) is an operation τ(f ) := ( {x xε{a}(0)}, x {a}(0) y ) is a collection (proposition).

57 Universes An extended realizability interpretation of sets and small propositions of mtt can be encoded using fix points. collections of ÎD 1 of (codes for) sets and small propositions of mtt: US and USP. f = [a] : 1 US(P) is an operation τ(f ) := ( {x xε{a}(0)}, x {a}(0) y ) is a collection (proposition). A set (small proposition) of ÎD 1 is a collection (proposition) of ÎD 1 of the form τ(f ) for f : 1 US(P).

58 first summary Obtain a commutative diagram in Cat

59 first summary Obtain a commutative diagram in Cat S P s C P

60 Contexts Define

61 Contexts Define a category of contexts of ÎD 1, Cont

62 Contexts Define a category of contexts of ÎD 1, Cont a functor Col : Cont op Cat

63 Contexts Define a category of contexts of ÎD 1, Cont a functor Col : Cont op Cat in such a way that if Γ is in Cont:

64 Contexts Define a category of contexts of ÎD 1, Cont a functor Col : Cont op Cat in such a way that if Γ is in Cont: A Col(Γ) [Γ, A] Cont

65 Contexts Define a category of contexts of ÎD 1, Cont a functor Col : Cont op Cat in such a way that if Γ is in Cont: A Col(Γ) [Γ, A] Cont f : A B in Col(Γ) f : pr [Γ,A] pr [Γ,B] in Cont/Γ

66 Contexts Define a category of contexts of ÎD 1, Cont a functor Col : Cont op Cat in such a way that if Γ is in Cont: A Col(Γ) [Γ, A] Cont f : A B in Col(Γ) f : pr [Γ,A] pr [Γ,B] in Cont/Γ where pr [Γ,A] is the projection from [Γ, A] to Γ.

67 Contexts Define a category of contexts of ÎD 1, Cont a functor Col : Cont op Cat in such a way that if Γ is in Cont: A Col(Γ) [Γ, A] Cont f : A B in Col(Γ) f : pr [Γ,A] pr [Γ,B] in Cont/Γ where pr [Γ,A] is the projection from [Γ, A] to Γ. We define appropriate functors Set, Prop, Prop s.

68 second summary Set Prop s Col Prop in Cat Contop

69 second summary Set Prop s Col Prop in Cat Contop USP cat US cat in Cat(C)

70 second summary Set Prop s Col Prop in Cat Contop USP cat US cat in Cat(C) S Set([ ]) Γ(US cat ) P s Prop s ([ ]) Γ(USP cat ) C Col([ ]) Cont P Prop([ ]) in Cat

71 Properties Col(Γ) and Set(Γ) are cartesian closed finitely complete with list objects and stable disjoint finite coproducts;

72 Properties Col(Γ) and Set(Γ) are cartesian closed finitely complete with list objects and stable disjoint finite coproducts; Col pr[γ,a] : Col(Γ) Col([Γ, A]) has left and right adjoints;

73 Properties Col(Γ) and Set(Γ) are cartesian closed finitely complete with list objects and stable disjoint finite coproducts; Col pr[γ,a] : Col(Γ) Col([Γ, A]) has left and right adjoints; Col(Γ) contains an universe object of small propositions in context;

74 Properties Col(Γ) and Set(Γ) are cartesian closed finitely complete with list objects and stable disjoint finite coproducts; Col pr[γ,a] : Col(Γ) Col([Γ, A]) has left and right adjoints; Col(Γ) contains an universe object of small propositions in context; Set pr[γ,a] : Set(Γ) Set([Γ, A]) has left and right adjoints if A Set([Γ]);

75 Properties Col(Γ) and Set(Γ) are cartesian closed finitely complete with list objects and stable disjoint finite coproducts; Col pr[γ,a] : Col(Γ) Col([Γ, A]) has left and right adjoints; Col(Γ) contains an universe object of small propositions in context; Set pr[γ,a] : Set(Γ) Set([Γ, A]) has left and right adjoints if A Set([Γ]); Prop is a first-order hyperdoctrine;

76 Properties Col(Γ) and Set(Γ) are cartesian closed finitely complete with list objects and stable disjoint finite coproducts; Col pr[γ,a] : Col(Γ) Col([Γ, A]) has left and right adjoints; Col(Γ) contains an universe object of small propositions in context; Set pr[γ,a] : Set(Γ) Set([Γ, A]) has left and right adjoints if A Set([Γ]); Prop is a first-order hyperdoctrine; Prop s is a doctrine with left and right adjoint for Prop s,pr[γ,a] : Prop s (Γ) Prop s ([Γ, A]) with A Set([Γ]);

77 (Partial) interpretation of (fully annotated!) syntax of mtt:

78 (Partial) interpretation of (fully annotated!) syntax of mtt: Contexts Γ objects I(Γ) of Cont;

79 (Partial) interpretation of (fully annotated!) syntax of mtt: Contexts Γ objects I(Γ) of Cont; Collections A in context Γ objects of Col(I(Γ));

80 (Partial) interpretation of (fully annotated!) syntax of mtt: Contexts Γ objects I(Γ) of Cont; Collections A in context Γ objects of Col(I(Γ)); Sets A in context Γ objects of Set(I(Γ));

81 (Partial) interpretation of (fully annotated!) syntax of mtt: Contexts Γ objects I(Γ) of Cont; Collections A in context Γ objects of Col(I(Γ)); Sets A in context Γ objects of Set(I(Γ)); Propositions A in context Γ objects of Prop(I(Γ));

82 (Partial) interpretation of (fully annotated!) syntax of mtt: Contexts Γ objects I(Γ) of Cont; Collections A in context Γ objects of Col(I(Γ)); Sets A in context Γ objects of Set(I(Γ)); Propositions A in context Γ objects of Prop(I(Γ)); Small propositions A in context Γ objects of Prop s (I(Γ));

83 (Partial) interpretation of (fully annotated!) syntax of mtt: Contexts Γ objects I(Γ) of Cont; Collections A in context Γ objects of Col(I(Γ)); Sets A in context Γ objects of Set(I(Γ)); Propositions A in context Γ objects of Prop(I(Γ)); Small propositions A in context Γ objects of Prop s (I(Γ)); Elements a in context Γ global elements in Col(I(Γ)).

84 (Partial) interpretation of (fully annotated!) syntax of mtt: Contexts Γ objects I(Γ) of Cont; Collections A in context Γ objects of Col(I(Γ)); Sets A in context Γ objects of Set(I(Γ)); Propositions A in context Γ objects of Prop(I(Γ)); Small propositions A in context Γ objects of Prop s (I(Γ)); Elements a in context Γ global elements in Col(I(Γ)).

85 Validity 1 Every judgment J of mtt for which mtt J is validated by the realizability model R

86 Validity 1 Every judgment J of mtt for which mtt J is validated by the realizability model R 2 R validates CT

87 Validity 1 Every judgment J of mtt for which mtt J is validated by the realizability model R 2 R validates CT 3 R validates AC N and AC!

88 Validity 1 Every judgment J of mtt for which mtt J is validated by the realizability model R 2 R validates CT 3 R validates AC N and AC! 4 from R, by elementary quotient completion (Maietti, Rosolini), model of emtt + CT

89 Validity 1 Every judgment J of mtt for which mtt J is validated by the realizability model R 2 R validates CT 3 R validates AC N and AC! 4 from R, by elementary quotient completion (Maietti, Rosolini), model of emtt + CT 5 validity of AC! is not preserved by completion

90 Validity 1 Every judgment J of mtt for which mtt J is validated by the realizability model R 2 R validates CT 3 R validates AC N and AC! 4 from R, by elementary quotient completion (Maietti, Rosolini), model of emtt + CT 5 validity of AC! is not preserved by completion

91 Work in progress Minimalist predicative version of tripos-to-topos construction

92 Work in progress Minimalist predicative version of tripos-to-topos construction Hyland s effective topos

93 Work in progress Minimalist predicative version of tripos-to-topos construction Hyland s effective topos realizability toposes

94 Work in progress Minimalist predicative version of tripos-to-topos construction Hyland s effective topos realizability toposes

95 References this work: M.E.Maietti, S.Maschio, A predicative realizability tripos for the Minimalist Foundations, in preparation.

96 References this work: M.E.Maietti, S.Maschio, A predicative realizability tripos for the Minimalist Foundations, in preparation. 1 Other papers on Minimalist Foundations 1 F.C., Toward a minimalist foundation for constructive mathematics, From Sets and Types to Topology and Analysis, M.E.Maietti, A minimalist two-level foundation for constructive mathematics, Annals of pure and applied logic, M.E.Maietti, G.Sambin Why topology in the Minimalist Foundation must be pointfree., Logic and Logical Philosophy, 2013.

97 References this work: M.E.Maietti, S.Maschio, A predicative realizability tripos for the Minimalist Foundations, in preparation. 1 Other papers on Minimalist Foundations 1 F.C., Toward a minimalist foundation for constructive mathematics, From Sets and Types to Topology and Analysis, M.E.Maietti, A minimalist two-level foundation for constructive mathematics, Annals of pure and applied logic, M.E.Maietti, G.Sambin Why topology in the Minimalist Foundation must be pointfree., Logic and Logical Philosophy, About Realizability Models for Type Theory and Minimalist Foundation 1 M. Beeson, Foundations of Constructive Mathematics, Springer-Verlag, M.E.Maietti, S.Maschio An extensional Kleene realizability semantics for the Minimalist Foundation, to appear in TYPES 14 post-proceedings, 2015.