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

1Source:

https://www.math.su.se/om-oss/nyheter/erik-palmgren-1963-2019-1.463835

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Introduction to Martin-L¨

• f Type Theory

Interpretation of Iterated Inductive Definitions

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Introduction to Martin-L¨

• f Type Theory

Introduction to Martin-L¨

• f Type Theory

Interpretation of Iterated Inductive Definitions

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Introduction to Martin-L¨

• f Type Theory

Martin-L¨

• f 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¨

• del’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.

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Introduction to Martin-L¨

• f 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

• ther 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¨

• f Type Theory

Π-Type

◮ Formation rule: A : Set B : A → Set Π A B : Set ◮ Introduction rule: x : A ⇒ t : B x λx.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 (λx.t) a = t[x := a] : B a

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Introduction to Martin-L¨

• f 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.

z′ y ′ y : B a z : B a′ f y = sup a′ f ′ f y ′ f ′ z′ sup a f (a : A, f : B a → W A B) If B a′′ empty we get a leaf f ′ z = sup a′′ f ′′

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Introduction to Martin-L¨

• f Type Theory

Kleene’s O

Example Kleene’s O, tree of height ω, Version in MLTT. KleeneOML

✿✿✿✿✿✿✿✿✿✿ := W A B, where A = {

∅, ∗, N} B ∅ = ∅ B ∗ = {∗} B N = N.

sup N f f 1 = sup ∗ g0 f 0 = sup ∅ efq f 2 = sup ∗ g1 g0 ∗ = sup ∅ efq g1 ∗ = sup ∗ g0 g0 ∗ = sup ∅ efq g1 ∗ = sup ∗ g0 g0 ∗ = sup ∅ efq g2 ∗ = sup ∗ g1 f 3 = sup ∗ g2 · · ·

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Introduction to Martin-L¨

• f Type Theory

KleeneOML,2

Example Kleene O2: ◮ KleeneOML,2

✿✿✿✿✿✿✿✿✿✿✿✿ := W A′ B′ where

◮ A′ = { ∅, ∗, N,

• KleeneO}

◮ B′ : A′ → Set B′ ∅ = ∅ B′ ∗ = {∗} B′ N = N B′

• KleeneO

= KleeneOML ◮ Therefore it’s a nested W-type.

◮ We can define ωck

1 : KleeneOML,2,

ωck

1

✿✿✿ := sup

• KleeneO embed

embed

✿✿✿✿✿✿ : KleeneOML → KleeneOML,2

embedding function. ◮ ωck

1 has height the supremum of the heights of all elements in

KleeneOML.

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Introduction to Martin-L¨

• f 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.

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Introduction to Martin-L¨

• f 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:

• N : U

T N = N a : U b : T a → U

• Π a b : U

(compare with A : Set b : A → Set Π A B : Set ) T( Π a b) = Π (T a) (T ◦ b) Similarly for other type formers (except for U).

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Interpretation of Iterated Inductive Definitions

Introduction to Martin-L¨

• f Type Theory

Interpretation of Iterated Inductive Definitions

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Interpretation of Iterated Inductive Definitions

Theory of Intuitionistic Inductive Definitions

◮ IDi 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 IA

✿ (written n ∈ IA)

◮ axioms expressing that IA is the least set closed under A: ∀n.A(IA, n) → n ∈ IA ∀n ∈ IA.A(B, n) → B(n) ∀n ∈ IA.B(n) where B(x) is any formula with distinguished variable x, which might make use of IA.

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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 IDi 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 if A(KleeneO, n) then n ∈ KleeneO

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Interpretation of Iterated Inductive Definitions

Kleene’s O as subset of N

{e}(1) ≃ 1, e0 {e}(2) ≃ 1, e1 {e}(3) ≃ 1, e2 {e}(0) ≃ e0 = 0, 0 e0 = 0, 0 e1 = 1, e0 e0 = 0, 0 e2 = 1, e1 e1 = 1, e0 e0 = 0, 0 · · · ω = 2, e

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Interpretation of Iterated Inductive Definitions

Theory of Intuitionistic Inductive Definitions

◮ IDi 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]) ψΩ1(ǫΩ1+1) ◮ Iterated inductive definitions were the topic of the famous monograph “BuFePoSi” [2].

2There is another notion of predicativity which gives limit Γ0.

J¨ ager calls theories between Γ0 and IDi

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

◮ IDi

n is the theory of n times iterated inductive definition.

◮ Allows predicates IA,k for k < n where IA,k can refer to IA′,k′ for k′ < k (positively and negatively).

◮ KleeneO2

✿✿✿✿✿✿✿✿ can be defined in IDi

2 as one inductive definition which

refers to KleeneO. ◮ Can be generalised to KleeneOn, definable in IDi

n.

◮ IDi

n = ψΩ1(ǫΩn+1) (e.g. [5]).

◮ IDi

<ω is the union of IDi n and has strength ψΩ1(Ωω) = |(Π1 1 − CA)0|.

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Interpretation of Iterated Inductive Definitions

Theory of transfinitely iterated intuitionistic inductive definitions

◮ We define the theory IDi

α

✿✿✿ of transfinitely iterated intuitionistic

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 IDi

α is given by

◮ for any predicate A(X, Y , β, n) strictly positive in X

◮ a binary predicate symbol n ∈ IA,β

✿✿✿✿✿✿

◮ a defined predicate IA,≺β

✿✿✿✿ :=

• γ≺β

{γ} × IA,γ

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Interpretation of Iterated Inductive Definitions

Theory of transfinitely iterated intuitionistic inductive definitions

◮ Axioms β ∈ OT A(IA,β, IA,≺β, β, n) n ∈ IA,β β ∈ OT ∀n ∈ IA,β.A(B, IA,≺β, β, n) → B(n) ∀n ∈ IA,β.B(n) ◮ Transfinite induction over OT.

◮ IDi

<α

✿✿✿✿ is the union of the theories IDi

β for β ≺ α.

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Interpretation of Iterated Inductive Definitions

Interpretation of Palmgren [4]

◮ Eric Palmgren was able to interpret IDi

<ǫ0 in

ML1W

✿✿✿✿✿✿ := MLTT + W + U

◮ This showed that the proof theoretic strength of the type theory in question is |ML1W| ≥ |IDi

<ǫ0| = |∆1 2 − CA| = ψΩ1(Ωǫ0)

◮ In our PhD thesis [6, 7] we showed that the strength is much bigger |ML1W| = ψΩ1(ΩI+ω) ◮ The proof required advanced well-ordering techniques due to Buchholz and Pohlers.3

3J¨

ager might have been involved as well - I haven’t investigated that yet. Our approach was based on the refined version by Buchholz, in draft version [1], see as well the book by Buchholz and Sch¨ utte [3]

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Interpretation of Iterated Inductive Definitions

Palmgren’s Results as a Solution to a revised Hilbert’s Program

◮ By Palmgren’s result, the strength of MLTT with W-type and one universe is > |(Π1

1 − CA)0|, which is the biggest of the big 5 systems

in reverse mathematics [9]. ◮ (Π1

1 − CA)0 allows to prove therefore most “real” mathematical

theories. ◮ ML1W proves its consistency. ◮ ML1W was designed to be a trustworthy theory (meaning explanations).4 ◮ If one trusts in this type theory, one can trust in the correctness of those proofs. ◮ Therefore Palmgren’s result gives a a first quite strong solution to a revised Hilbert’s program.

4Trustworthiness is subject to a philosophical debate Anton Setzer Erik Palmgren and Hilbert’s Program? 22/ 30

Interpretation of Iterated Inductive Definitions

Sharpening the Bounds of Palmgren

◮ When revisiting Palmgren’s proof one sees that he didn’t use the full power of ML1W.

◮ We can restrict W-types to elements of the universe. So we define (W a b) only for a : U and b : T a → U. ◮ We can restrict induction over W-types to elements of the universe. ◮ Let the resulting theory be called ML1W−

✿✿✿✿✿✿✿.

◮ Subject to working out the full details of the proof we obtain the following result [8]:

◮ The interpretation of IDi

<ǫ0 by Palmgren can be carried out as well in

ML1W−. ◮ ML1W− can be interpreted in IDi

<ǫ0

◮ Therefore |ML1W−| = |IDi

<ǫ0| = ψΩ1(Ωǫ0).

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Interpretation of Iterated Inductive Definitions

Conclusion

◮ Palmgren showed that IDi

<ǫ0 can be interpreted in ML1W.

◮ Therefore ML1W shows the consistency of (Π1

1 − CA)0 sufficient to

carry out most real mathematical proofs. ◮ Therefore Palmgren’s result gives an answer to a revised Hilbert’s program. ◮ The result can be sharped to determine the precise strength of a weaker theory ML1W−.

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Interpretation of Iterated Inductive Definitions

(Optional Slide) Proof of Palmgren

◮ Strictly positive inductive definitions give rise to a monotone operator Γ : P(N) → P(N) where P(N) = N → U. ◮ For a strictly positive inductive definition one can “collect” all the sets, ∀-quantifiers in its definition are ranging over. ◮ Now one defines a W-type which has as branching degree all those sets. ◮ If we iterate the operator Γ transfinitely over the W-type, one obtains the least fixed point of Γ which one can use to interpret an inductive definition. ◮ By “Gentzen’s trick” one obtains transfinite induction up to < ǫ0 over types, and can use it to get iterated inductive definitions up to α for any α < ǫ0.

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Interpretation of Iterated Inductive Definitions

(Optional Slide) Further Result by Palmgren

◮ Erik Palmgren showed as well in [4] that if one replaces the W-type in type theory by finitely iterated versions of Aczel’s V

◮ Used by Aczel to interpret constructive set theory CZF in type theory

• ne obtains the strength |IDi

<ω| = ψΩ1(Ωω) = |(Π1 1 − CA)0|

(as noted before)

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Interpretation of Iterated Inductive Definitions

Bibliography I

• W. Buchholz.

Wellordering proofs for systems of fundamental sequences. Draft, M¨ unchen, 1990.

• W. Buchholz, S. Feferman, W. Pohlers, W. Sieg, S. Feferman, and
• W. Sieg, editors.

Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies, volume 897 of Lecture Notes in

• Mathematics. Springer Berlin / Heidelberg, 1981.

10.1007/BFb0091895.

• W. Buchholz and K. Sch¨

utte. Proof Theory of Impredicative Subsystems of Analysis. Studies in Proof Theory, Monographs, Vol 2. Bibliopolis, Naples, 1989.

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Interpretation of Iterated Inductive Definitions

Bibliography II

• E. Palmgren.

Type-theoretic interpretation of iterated, strictly positive inductive definitions. Archive for Mathematical Logic, 32:75–99, 1992.

• W. Pohlers.

Chapter 4: Subsystems of set theory and second order number theory. In S. R. Buss, editor, Handbook of Proof Theory, volume 137 of Studies in Logic and the Foundations of Mathematics, pages 209 –

• 335. Elsevier, 1998.

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Interpretation of Iterated Inductive Definitions

Bibliography III

• A. Setzer.

Proof theoretical strength of Martin-L¨

• f Type Theory with W-type

and one universe. PhD thesis, Mathematisches Institut, Universit¨ at M¨ unchen, Munich, Germany, 1993. Available from http://www.cs.swan.ac.uk/~csetzer/articles/weor0.pdf.

• A. Setzer.

Well-ordering proofs for Martin-L¨

• f type theory.

Annals of Pure and Applied Logic, 92:113 – 159, 1998. https://doi.org/10.1016/S0168-0072(97)00078-X, http://www.cs.swan.ac.uk/~csetzer/articles/2papdiss.pdf.

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Interpretation of Iterated Inductive Definitions

Bibliography IV

• A. Setzer.

Did Palmgren solve a revised Hilbert’s program?, March 2020. In Abstracts of Types 2020. https://types2020.di.unito.it/ abstracts/BookOfAbstractsTYPES2020.pdf.

• S. G. Simpson and S. S. G. Simpson.

Subsystems of second order arithmetic, volume 1. Cambridge University Press, 2nd edition, 2009.

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