Did Erik Palmgren Solve a Revised Hilberts Program? Anton Setzer - PowerPoint PPT Presentation

Did Erik Palmgren Solve a Revised Hilbert’s Program? Anton Setzer Dept. of Computer Science, Swansea University, UK BCTCS 2020, 8 April 2020 In Living Memory of Erik Palmgren Anton Setzer Erik Palmgren and Hilbert’s Program? 1/ 30

Erik Palmgren, 1963 - 2019 1 1 Source: https://www.math.su.se/om-oss/nyheter/erik-palmgren-1963-2019-1.463835 Anton Setzer Erik Palmgren and Hilbert’s Program? 2/ 30

Introduction to Martin-L¨ of Type Theory Interpretation of Iterated Inductive Definitions Anton Setzer Erik Palmgren and Hilbert’s Program? 3/ 30

Introduction to Martin-L¨ of Type Theory Introduction to Martin-L¨ of Type Theory Interpretation of Iterated Inductive Definitions Anton Setzer Erik Palmgren and Hilbert’s Program? 4/ 30

Introduction to Martin-L¨ of Type Theory Martin-L¨ of Type Theory Martin-Löf ✿✿✿✿✿ Type Theory ( ✿✿✿✿✿✿ MLTT ) is a type theory for formalising ◮ ✿✿✿✿✿✿✿✿✿✿✿ ✿✿✿✿✿✿✿✿ constructive mathematics. ◮ It is designed in such a way that one has – as far as possible – a direct insight into the validity of its judgements . ◮ As a response to the failure of the original Hilbert’s program due to G¨ odel’s 2nd Incompleteness Theorem. ◮ MLTT is as well the basis for the theoretical basis for the interactive theorem prover and dependently typed programming language Agda . Anton Setzer Erik Palmgren and Hilbert’s Program? 5/ 30

Introduction to Martin-L¨ of Type Theory Dependent Type Theory ◮ Simple Type Theory has non dependent types, the main ones being A × B A → B ◮ Dependent Type Theory allows types to dependent on elements of other types. ◮ One of the origins is the interpretation of the ∀ -quantifier. ◮ In BHK interpretation of logical connectives, a proof of ∀ x : A . B x ◮ is a function that ◮ maps an element x : A to a proof of B x . ◮ So proofs are elements of type Π A B . ◮ Π A B ✿✿✿✿✿ = type of dependent functions, which map x : A to an element of B x . ◮ Remark: Set ✿✿✿ in MLTT is what is usually called “ Type ”. Anton Setzer Erik Palmgren and Hilbert’s Program? 6/ 30

Introduction to Martin-L¨ of Type Theory Π-Type ◮ Formation rule: A : Set B : A → Set Π A B : Set ◮ Introduction rule: x : A ⇒ t : B x λ x . t : Π A B ◮ Elimination rule: f : Π A B a : A Ap f a : B a ◮ Equality rule: x : A ⇒ t : B x a : A Ap ( λ x . t ) a = t [ x := a ] : B a Anton Setzer Erik Palmgren and Hilbert’s Program? 7/ 30

Introduction to Martin-L¨ of Type Theory W -Type Assume A : Set , B : A → Set . W A B ✿✿✿✿✿✿ is the type of well-founded recursive trees with branching degrees ( B a ) a : A . If B a ′′ empty we get a leaf f ′ z = sup a ′′ f ′′ f ′ z ′ z : B a ′ z ′ f y = sup a ′ f ′ f y ′ y ′ y : B a sup a f ( a : A , f : B a → W A B ) Anton Setzer Erik Palmgren and Hilbert’s Program? 8/ 30

Introduction to Martin-L¨ of Type Theory Kleene’s O Example Kleene’s O, tree of height ω , Version in MLTT. ✿✿✿✿✿✿✿✿✿✿ := W A B , where A = { � ∗ , � KleeneO ML ∅ , � N } B � B � B � N = N . ∅ = ∅ ∗ = {∗} g 0 ∗ = sup � ∅ efq g 0 ∗ = sup � ∅ efq g 1 ∗ = sup � ∗ g 0 g 0 ∗ = sup � ∅ efq g 1 ∗ = sup � ∗ g 0 g 2 ∗ = sup � ∗ g 1 f 0 = sup � ∅ efq f 1 = sup � ∗ g 0 f 2 = sup � ∗ g 1 f 3 = sup � ∗ g 2 · · · sup � N f Anton Setzer Erik Palmgren and Hilbert’s Program? 9/ 30

Introduction to Martin-L¨ of Type Theory KleeneO ML , 2 Example Kleene O 2 : ✿✿✿✿✿✿✿✿✿✿✿✿ := W A ′ B ′ where ◮ KleeneO ML , 2 ◮ A ′ = { � � ∗ , � ∅ , � N , KleeneO } ◮ B ′ : A ′ → Set B ′ � ∅ = ∅ B ′ � ∗ = {∗} B ′ � N = N � B ′ = KleeneO KleeneO ML ◮ Therefore it’s a nested W -type . ◮ We can define ω ck 1 : KleeneO ML , 2 , � ω ck ✿✿✿ := sup KleeneO embed 1 embed ✿✿✿✿✿✿ : KleeneO ML → KleeneO ML , 2 embedding function. ◮ ω ck 1 has height the supremum of the heights of all elements in KleeneO ML . Anton Setzer Erik Palmgren and Hilbert’s Program? 10/ 30

Introduction to Martin-L¨ of Type Theory The W -Type ◮ Formation rule: A : Set B : A → Set W A B : Set ◮ Introduction rule: a : A b : B a → W A B sup a b : W A B ◮ Elimination and Equality Rules: Induction over trees. Anton Setzer Erik Palmgren and Hilbert’s Program? 11/ 30

Introduction to Martin-L¨ of Type Theory Universes ◮ A universe is a family of sets ◮ Given by ◮ an set U : Set of codes for sets, ◮ a decoding function T : U → Set . ◮ Formation rules: U : Set T : U → Set ◮ Introduction and Equality rules: � T � N : U N = N b : T a → U a : U (compare with A : Set b : A → Set ) � Π A B : Set Π a b : U T ( � Π a b ) = Π ( T a ) ( T ◦ b ) Similarly for other type formers (except for U ). Anton Setzer Erik Palmgren and Hilbert’s Program? 12/ 30

Interpretation of Iterated Inductive Definitions Introduction to Martin-L¨ of Type Theory Interpretation of Iterated Inductive Definitions Anton Setzer Erik Palmgren and Hilbert’s Program? 13/ 30

Interpretation of Iterated Inductive Definitions Theory of Intuitionistic Inductive Definitions ◮ ID i is the theory of intuitionistic inductive definitions given by ◮ The language and theory HA of Heyting Arithmetic, ◮ for formulas A ( X , y ) strictly positive in X ◮ a predicate I A ✿ (written n ∈ I A ) ◮ axioms expressing that I A is the least set closed under A : ∀ n . A ( I A , n ) → n ∈ I A ∀ n ∈ I A . A ( B , n ) → B ( n ) ∀ n ∈ I A . B ( n ) where B ( x ) is any formula with distinguished variable x , which might make use of I A . Anton Setzer Erik Palmgren and Hilbert’s Program? 14/ 30

Interpretation of Iterated Inductive Definitions Example: Inductive Definition of Kleene’s O ◮ KleeneO ✿✿✿✿✿✿✿✿ (Kleene’s O as a set of natural numbers) can be defined inductively by ◮ � 0 , 0 � ∈ KleeneO . ◮ If e ∈ KleeneO then � 1 , e � ∈ KleeneO ◮ If ∀ n ∈ N . { e } ( n ) ∈ KleeneO , then � 2 , e � ∈ KleeneO . ◮ Definable in ID i using A ( X , n ) ✿✿✿✿✿✿✿ := ( n = � 0 , 0 � ∨ ( ∃ m . n = � 1 , m � ∧ m ∈ X ) ∨ ( ∃ e . n = � 1 , e � ∧ ∀ m . ∃ k . { e } ( m ) ≃ k ∧ k ∈ X )) ◮ So the above definition is equvialent to the inductive definition A ( KleeneO , n ) n ∈ KleeneO if then Anton Setzer Erik Palmgren and Hilbert’s Program? 15/ 30

Interpretation of Iterated Inductive Definitions Kleene’s O as subset of N e 0 = � 0 , 0 � e 0 = � 0 , 0 � e 1 = � 1 , e 0 � e 1 = � 1 , e 0 � e 0 = � 0 , 0 � e 2 = � 1 , e 1 � { e } (0) ≃ e 0 = � 0 , 0 � { e } (1) ≃ � 1 , e 0 � { e } (2) ≃ � 1 , e 1 � { e } (3) ≃ � 1 , e 2 � · · · ω = � 2 , e � Anton Setzer Erik Palmgren and Hilbert’s Program? 16/ 30

Interpretation of Iterated Inductive Definitions Theory of Intuitionistic Inductive Definitions ◮ ID i is the smallest (in a proof theoretic sense) fully impredicative theory studied in proof theory. 2 ◮ It’s strength is the Bachmann Howard Ordinal, in modern notation (e.g. [5]) ψ Ω 1 ( ǫ Ω 1 +1 ) ◮ Iterated inductive definitions were the topic of the famous monograph “BuFePoSi” [2]. 2 There is another notion of predicativity which gives limit Γ 0 . ager calls theories between Γ 0 and ID i J¨ 1 “meta-predicative”. Anton Setzer Erik Palmgren and Hilbert’s Program? 17/ 30

Interpretation of Iterated Inductive Definitions Theory of Finitely Iterated Intuitionistic Inductive Definitions ◮ ID i n is the theory of n times iterated inductive definition. ◮ Allows predicates I A , k for k < n where I A , k can refer to I A ′ , k ′ for k ′ < k (positively and negatively). ◮ KleeneO 2 ✿✿✿✿✿✿✿✿ can be defined in ID i 2 as one inductive definition which refers to KleeneO . ◮ Can be generalised to KleeneO n , definable in ID i n . ◮ ID i n = ψ Ω 1 ( ǫ Ω n +1 ) (e.g. [5]). ◮ ID i <ω is the union of ID i n and has strength ψ Ω 1 (Ω ω ) = | (Π 1 1 − CA ) 0 | . Anton Setzer Erik Palmgren and Hilbert’s Program? 18/ 30

Interpretation of Iterated Inductive Definitions Theory of transfinitely iterated intuitionistic inductive definitions ◮ We define the theory ID i ✿✿✿ of transfinitely iterated intuitionistic α inductive definitions: ◮ Fix an ordinal notation system ( OT , ≺ ) ✿✿✿✿✿✿✿ of order type α , i.e. ◮ OT ⊆ N primitive recursive, ◮ ≺ primitive recursive binary relation on OT , ◮ ( OT , ≺ ) well founded of order type α . ◮ β, γ, . . . refer to elements of OT . ◮ Language of ID i α is given by ◮ for any predicate A ( X , Y , β, n ) strictly positive in X ◮ a binary predicate symbol n ∈ I A ,β ✿✿✿✿✿✿ ◮ a defined predicate � I A , ≺ β ✿✿✿✿ := { γ } × I A ,γ γ ≺ β Anton Setzer Erik Palmgren and Hilbert’s Program? 19/ 30