Ordinal Strength of Logic-Enriched Type Theories

Ordinal Strength of Logic-Enriched Type Theories Robin Adams Royal Holloway, University of London 27 March 2012 Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 1 / 27

Introduction Type theories are formal languages in which both algorithms and proofs may be expressed. Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 2 / 27

Introduction Type theories are formal languages in which both algorithms and proofs may be expressed. There should therefore be applications to proof complexity. Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 2 / 27

Introduction Type theories are formal languages in which both algorithms and proofs may be expressed. There should therefore be applications to proof complexity. The type theory and mainstream logical communities have been very separate. One reason is it is difficult to translate directly between a type theory and a mainstream logic (syntax or semantics). Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 2 / 27

Introduction Type theories are formal languages in which both algorithms and proofs may be expressed. There should therefore be applications to proof complexity. The type theory and mainstream logical communities have been very separate. One reason is it is difficult to translate directly between a type theory and a mainstream logic (syntax or semantics). Logic-enriched type theories (LTTs) may be the bridge we need. Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 2 / 27

Introduction Type theories are formal languages in which both algorithms and proofs may be expressed. There should therefore be applications to proof complexity. The type theory and mainstream logical communities have been very separate. One reason is it is difficult to translate directly between a type theory and a mainstream logic (syntax or semantics). Logic-enriched type theories (LTTs) may be the bridge we need. I will describe my work so far on LTTs, and some problems I believe LTTs may be able to help with. Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 2 / 27

1 Introduction to Type Theory and LTTs 2 Weyl s School of Predicativism Proof Technique for Conservativity Ordinal Strength of Weyl s Foundation 3 Potential Applications Ordinal Strength of Theories Application to Finite Model Theory 4 Conclusion Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 3 / 27

Type Theory A type theory is a formal system that deals with judgements of the for a is an object of type A a and b are equal objects of type A Examples: x 1 : A 1,..., x n : A n a : A x 1 : A 1,..., x n : A n a = b : A 0 : N x : N x, 7 : N N λx : N.x + 5 : N N f : N N f (5) : N λx : A.b is the function f with domain A such that f (x) = b for all x : A. Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 4 / 27

Type Theory Objects Types Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 5 / 27

Type Theory Objects Types Natural numbers 0, s0, ss0,... N Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 5 / 27

Type Theory Objects Types Natural numbers 0, s0, ss0,... N Pairs a, b A B Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 5 / 27

Type Theory Objects Types Natural numbers 0, s0, ss0,... N Pairs a, b A B Functions λx.b[x] A B Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 5 / 27

Type Theory Objects Types Natural numbers 0, s0, ss0,... N Pairs a, b A B Functions λx.b[x] A B Variables x A Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 5 / 27

Type Theory Objects Types Natural numbers 0, s0, ss0,... N Pairs a, b A B Functions λx.b[x] A B Variables x A Vectors : Vec(A, 0) Vec(A, n) l :: a : Vec(A, sn) Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 5 / 27

Type Theory Objects Types Natural numbers 0, s0, ss0,... N Pairs a, b A B Functions λx.b[x] A B Variables x A Vectors : Vec(A, 0) Vec(A, n) l :: a : Vec(A, sn) a, b Σx : A.B[x] Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 5 / 27

Type Theory Objects Types Natural numbers 0, s0, ss0,... N Pairs a, b A B Functions λx.b[x] A B Variables x A Vectors : Vec(A, 0) Vec(A, n) l :: a : Vec(A, sn) a, b Σx : A.B[x] λx.b[x] Πx : A.B[x] Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 5 / 27

Type Theory Objects Types Natural numbers 0, s0, ss0,... N Pairs a, b A B Functions λx.b[x] A B Variables x A Vectors : Vec(A, 0) Vec(A, n) l :: a : Vec(A, sn) a, b Σx : A.B[x] λx.b[x] Πx : A.B[x] one object if a = b I (A, a, b) no objects if a b Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 5 / 27

Type Theory Objects Types Natural numbers 0, s0, ss0,... N Pairs a, b A B Functions λx.b[x] A B Variables x A Vectors : Vec(A, 0) Vec(A, n) l :: a : Vec(A, sn) a, b Σx : A.B[x] λx.b[x] Πx : A.B[x] one object if a = b I (A, a, b) no objects if a b Universe N, N N, N N,... U Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 5 / 27

Rules in Type Theory Functions Pairs f : A B f (a) : B a : A b : B λx : A.b : B a : A b : B (a, b) : A B a : A B a : A B a.1 : A a.2 : B Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 6 / 27

Wait A Minute... Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 7 / 27

Wait A Minute... Term M ::= x λx.m MM Type A ::= α A A Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 7 / 27

Wait A Minute... Term M ::= x λx.m MM Type A ::= α A A x 1 :A 1,..., x n :A n x i :A i Γ F :A B Γ M :A Γ FM :B Γ, x :A M :B Γ λx.m :A B Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 7 / 27

Wait A Minute... Term M ::= x λx.m MM Type A ::= α A A x 1 :A 1,..., x n :A n x i :A i Γ F :A B Γ M :A Γ FM :B Γ, x :A M :B Γ λx.m :A B Curry (1958) noticed something strange... Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 7 / 27

Wait A Minute... Term M ::= x λx.m MM Type A ::= α A A A 1,..., A n A i Γ B Γ A B Γ A Γ, A B Γ A B Curry (1958) noticed something strange... Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 7 / 27

The Plot Thickens... Howard (1969): Term M ::= x λx.m MM Type A ::= α A A x 1 : A 1,..., x n : A n x i : A i Γ F : A B Γ M : A Γ FM : B Γ, x : A M : B Γ λx.m : A B Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 8 / 27

The Plot Thickens... Howard (1969): Term M ::= x λx.m MM (M, M) π 1 (M) π 2 (M) Type A ::= α A A A A x 1 : A 1,..., x n : A n x i : A i Γ F : A B Γ M : A Γ FM : B Γ, x : A M : B Γ λx.m : A B Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 8 / 27

The Plot Thickens... Howard (1969): Term M ::= x λx.m MM (M, M) π 1 (M) π 2 (M) Type A ::= α A A A A x 1 : A 1,..., x n : A n x i : A i Γ F : A B Γ M : A Γ FM : B Γ, x : A M : B Γ λx.m : A B Γ M : A Γ N : B Γ M : A B Γ (M, N) : A B Γ π 1 (M) : A Γ M : A B Γ π 2 (M) : B Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 8 / 27

The Plot Thickens... Howard (1969): Type A ::= α A A A A x 1 : A 1,..., x n : A n x i : A i Γ F : A B Γ M : A Γ FM : B Γ, x : A M : B Γ λx.m : A B Γ M : A Γ N : B Γ M : A B Γ (M, N) : A B Γ π 1 (M) : A Γ M : A B Γ π 2 (M) : B Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 8 / 27

The Plot Thickens... Howard (1969): Type A ::= α A A A A A 1,..., A n A i Γ B Γ A B Γ A Γ, A B Γ A B Γ A Γ B Γ A B Γ A B Γ A B Γ A Γ B Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 8 / 27

The Plot Thickens... Howard (1969): Type A ::= α A A A A A 1,..., A n A i Γ B Γ A B Γ A Γ, A B Γ A B Γ A Γ A B Γ A Γ B Γ A B Γ A B Γ B Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 9 / 27

Propositions-as-Types Type theories can be given a second interpretation read a : A as a is a proof of the proposition A When read this way: A B is the proposition A and B A B is the proposition A implies B Πx : A.B is the proposition for all x : A, B Σx : A.B is the proposition there exists x : A such that B I (A, a, b) is the proposition a equals b The fact that the formal rules for typing and logic are identical is known as the Curry-Howard isomorphism. Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 10 / 27

Relating Type Theories and Mainstream Logics In logic, we write down axioms, then discover which objects these allow us to define (i.e. which objects we can prove exist / which objects exist in every model). In type theory, we write down types (mostly inductively defined collections) based on which objects we wish to have, then discover what logical principles these allow us. Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 11 / 27

Relating Type Theories and Mainstream Logics In logic, we write down axioms, then discover which objects these allow us to define (i.e. which objects we can prove exist / which objects exist in every model). In type theory, we write down types (mostly inductively defined collections) based on which objects we wish to have, then discover what logical principles these allow us. The two worlds are difficult to relate directly. Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 11 / 27

Relating ML 1 V and CZF History Aczel and Gambino [GA06] investigated the relationship between Constructive ZF set theory (CZF) and the type theory ML 1 V. Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 12 / 27

Relating ML 1 V and CZF History Aczel and Gambino [GA06] investigated the relationship between Constructive ZF set theory (CZF) and the type theory ML 1 V. They discovered it helps to introduce an intermediate system, a logic-enriched type theory (LTT). Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 12 / 27

Logic-Enriched Type Theories A logic-enriched type theory (LTT) consists of: a type theory a separate set of formulas a set of rules that determine which formulas are provable. Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 13 / 27

Logic-Enriched Type Theories A logic-enriched type theory (LTT) consists of: a type theory a separate set of formulas a set of rules that determine which formulas are provable. It thus has two worlds the logical world and the type theory world. Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 13 / 27

Logic-Enriched Type Theories A logic-enriched type theory (LTT) consists of: a type theory a separate set of formulas a set of rules that determine which formulas are provable. It thus has two worlds the logical world and the type theory world. These two worlds interact but can be modified separately. Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 13 / 27

Predicativism A definition is impredicative if it involves a certain kind of circularity: quantifying over all sets when defining a set quantifying over all real numbers when defining a real number. Example: The definition of the least upper bound of a set of reals is impredicative, as it involves quantifying over all real numbers. Predicativism is the view that impredicative definitions are illegitimate. How can we restrict our methods of definition so that impredicative definitions are ruled out? Weyl s solution (1914): Divide mathematical objects into categories. Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 14 / 27

Predicativism A definition is impredicative if it involves a certain kind of circularity: quantifying over all sets when defining a set quantifying over all real numbers when defining a real number. Example: The definition of the least upper bound of a set of reals is impredicative, as it involves quantifying over all real numbers. Predicativism is the view that impredicative definitions are illegitimate. How can we restrict our methods of definition so that impredicative definitions are ruled out? Weyl s solution (1914): Divide mathematical objects into categories. Divide categories into basic and ideal categories. Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 14 / 27

Predicativism A definition is impredicative if it involves a certain kind of circularity: quantifying over all sets when defining a set quantifying over all real numbers when defining a real number. Example: The definition of the least upper bound of a set of reals is impredicative, as it involves quantifying over all real numbers. Predicativism is the view that impredicative definitions are illegitimate. How can we restrict our methods of definition so that impredicative definitions are ruled out? Weyl s solution (1914): Divide mathematical objects into categories. Divide categories into basic and ideal categories. Natural numbers form a basic category. Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 14 / 27

Predicativism A definition is impredicative if it involves a certain kind of circularity: quantifying over all sets when defining a set quantifying over all real numbers when defining a real number. Example: The definition of the least upper bound of a set of reals is impredicative, as it involves quantifying over all real numbers. Predicativism is the view that impredicative definitions are illegitimate. How can we restrict our methods of definition so that impredicative definitions are ruled out? Weyl s solution (1914): Divide mathematical objects into categories. Divide categories into basic and ideal categories. Natural numbers form a basic category. For any category A, the sets of As form an ideal category. Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 14 / 27

Predicativism A definition is impredicative if it involves a certain kind of circularity: quantifying over all sets when defining a set quantifying over all real numbers when defining a real number. Example: The definition of the least upper bound of a set of reals is impredicative, as it involves quantifying over all real numbers. Predicativism is the view that impredicative definitions are illegitimate. How can we restrict our methods of definition so that impredicative definitions are ruled out? Weyl s solution (1914): Divide mathematical objects into categories. Divide categories into basic and ideal categories. Natural numbers form a basic category. For any category A, the sets of As form an ideal category. When defining a set, we may only quantify over the basic categories. Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 14 / 27

ACA 0 Weyl s Foundation as a System of Predicate Logic ACA 0 is a modern formulation of Weyl s system (Feferman). Two sorts: natural numbers and sets of natural numbers (second-order language). Axioms: Peano s axioms Restricted induction φ[0] x(φ[x] φ[x ]) xφ[x] for φ[x] arithmetic Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 15 / 27

ACA 0 Weyl s Foundation as a System of Predicate Logic ACA 0 is a modern formulation of Weyl s system (Feferman). Two sorts: natural numbers and sets of natural numbers (second-order language). Axioms: Peano s axioms Restricted induction Arithmetic Comprehension Axiom φ[0] x(φ[x] φ[x ]) xφ[x] for φ[x] arithmetic {x : φ[x]} exists for φ[x] arithmetic Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 15 / 27

ACA 0 Weyl s Foundation as a System of Predicate Logic ACA 0 is a modern formulation of Weyl s system (Feferman). Two sorts: natural numbers and sets of natural numbers (second-order language). Axioms: Peano s axioms Restricted induction Arithmetic Comprehension Axiom Conservative extension of PA. φ[0] x(φ[x] φ[x ]) xφ[x] for φ[x] arithmetic {x : φ[x]} exists for φ[x] arithmetic Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 15 / 27

ACA 0 Weyl s Foundation as a System of Predicate Logic ACA 0 is a modern formulation of Weyl s system (Feferman). Two sorts: natural numbers and sets of natural numbers (second-order language). Axioms: Peano s axioms Restricted induction Arithmetic Comprehension Axiom Conservative extension of PA. Shortcomings: Weyl uses sets of sets definition of sets by recursion full induction φ[0] x(φ[x] φ[x ]) xφ[x] for φ[x] arithmetic {x : φ[x]} exists for φ[x] arithmetic Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 15 / 27

LTT W Weyl s Foundation as a Logic-Enriched Type Theory Define an LTT named LTT W [AL10b]: Type Theory World Universe U (objects are the basic categories) Logical World Universe prop (objects are the arithmetic formulas) N Set(A) =,,,, Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 16 / 27

LTT W Weyl s Foundation as a Logic-Enriched Type Theory Define an LTT named LTT W [AL10b]: Type Theory World Universe U (objects are the basic categories) Logical World Universe prop (objects are the arithmetic formulas) N Set(A) =,,,, All Weyl s results can be formalised in LTT W checked by proof assistant Plastic. Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 16 / 27

ACA 0 is Embeddable in LTT W Formulas of ACA 0 Formulas of LTT W Define a mapping Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 17 / 27

ACA 0 is Embeddable in LTT W Define a mapping Formulas of ACA 0 Formulas of LTT W = =.. x x : N x x : N X X : Set(N) X X : Set(N) Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 17 / 27

ACA 0 is Embeddable in LTT W Formulas of ACA 0 Formulas of LTT W = = Define a mapping.. x x : N x x : N X X : Set(N) X X : Set(N) But LTT W goes further: has types Set(Set(N)),... allows definition by recursion in Set(A) allows full induction Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 17 / 27

Subsystems of LTT W In LTT W, we can eliminate over any type. Let us define the following subsystems of LTT W : Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 18 / 27

Subsystems of LTT W In LTT W, we can eliminate over any type. Let us define the following subsystems of LTT W : LTT 0 restrict E N to the members of U and induction to prop. LTT W 1 restrict E N to the members of U. LTT W 2 restrict E N to A and PA (A : U) LTT W 3 restrict E N to A, PA, PPA (A : U) Theorem ([AL10a]) ACA 0 can be conservatively embedded in LTT 0. ACA can be conservatively embedded in LTT 1 W. ACA + 0 can be embedded in LTT W 2 ( Conjecture: This is not conservative.) Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 18 / 27

Subsystems of LTT W In LTT W, we can eliminate over any type. Let us define the following subsystems of LTT W : LTT 0 restrict E N to the members of U and induction to prop. LTT W 1 restrict E N to the members of U. LTT W 2 restrict E N to A and PA (A : U) LTT W 3 restrict E N to A, PA, PPA (A : U) Theorem ([AL10a]) ACA 0 can be conservatively embedded in LTT 0. ACA can be conservatively embedded in LTT 1 W. ACA + 0 can be embedded in LTT W 2 ( Conjecture: This is not conservative.) Conjecture LTT W n+1 is never conservative over LTT W n. Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 18 / 27

Conservativity of ACA 0 over PA We interpret the formulas of ACA 0 as being statements about the syntax and theorems of PA. Define a valuation to be a mapping of: first-order variables to terms of PA; second-order variables to formulas of PA. Define what it means for a valuation v to satisfy a formula φ thus: If φ is arithmetic, then v = φ iff v(φ) is a theorem of PA.... A formula is true iff it is satisfied by every valuation. Prove: Every theorem of ACA 0 is true. Every first-order formula that is true is a theorem of PA. Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 19 / 27

Ordinal Strength of a Type Theory Definition A type theory is consistent iff there is no term M such that M : in derivable We can code the terms, types,... of a type theory with Gödel numbers, and represent the relation J is derivable in PRA. Definition The ordinal strength of a type theory T is the least ordinal α such that PRA + TI (α) M( T M : ) Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 20 / 27

Proof Techniques for Ordinal Strength To show that a predicate logic theory T has ordinal strength α: Prove that T α by showing that T TI (β) for all β < α (Easy). Prove that T α by proving Con(T ) in a theory of strength α (Hard). To show that a type theory T has ordinal strength α: Prove that T α by constructing in T a well-ordering of length α (Hard). Prove that T α by constructing a model of T in a variant of KP set theory of strength α (Easy). Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 21 / 27

Result LTT W has ordinal φ ɛ0 (0) Setzer and I recently proved that LTT W has strength φ ɛ0 (0), the same strength as ACA, IDˆ 1 and ML 1. Sketch Proof Extend LTT W with a type AE that represents the ordinals below Γ 0 : 0 : AE α : AE β : AE α + ω β : AE α : AE β : AE φ α (β) : AE This is a conservative extension (proof: members of AE can be coded as natural numbers). Definition (Progressive) X : PAE is progressive iff α( β < α.β X ) α X α : AE is accessible iff it is in every progressive set. Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 22 / 27

Result LTT W has ordinal φ ɛ0 (0) Define a lens for a function φ : AE AE to be a function Φ : PAE PAE such that: if X is progressive then Φ(X ) is progressive; if X is progressive then Φ(X ) φ 1 (X ). Prove that: 1 If there is a lens for φ then the accessible ordinals are closed under φ. 2 If there is a lens for φ then there is a lens for φ n and φ ω. It follows that, for all α < ɛ 0, LTT W proves that there is a lens for φ α. Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 23 / 27

Result LTT W has ordinal φ ɛ0 (0) Define a lens for a function φ : AE AE to be a function Φ : PAE PAE such that: if X is progressive then Φ(X ) is progressive; if X is progressive then Φ(X ) φ 1 (X ). Prove that: 1 If there is a lens for φ then the accessible ordinals are closed under φ. 2 If there is a lens for φ then there is a lens for φ n and φ ω. It follows that, for all α < ɛ 0, LTT W proves that there is a lens for φ α. As soon as we had the result, we saw that there is an easier way: ˆ ID 1 LTT W ML 1 Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 23 / 27

Idea Can we relate a predicate logic and a type theory, via an LTT, so that we know they have the same ordinal strength? L LTT T We could then prove a lower bound for L and an upper bound for T. Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 24 / 27

Application to Finite Model Theory Work in Progress For a set K of ordered structures: membership of K is in P iff K is axiomatised by a sentence of FO(IFP); membership of K is in NP iff K is axiomatised by a Σ 1 -sentence. We can make prop the set of formulas of FO(IFP) by adding IFP : (A : U)(PA PA) PA PA We can also make prop the set of Σ 1 sentences, by closing it under =,,,, There is ongoing work in type theory on fixed points for type constructors: Could LTTs provide a bridge between the model theory of FO(IFP) and type theories with fixed points? Perhaps providing a proof theory for FO(IFP)? Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 25 / 27

Conclusion LTTs might possibly be a useful tool for bridging type theories and mainstream logics. Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 26 / 27

Conclusion LTTs might possibly be a useful tool for bridging type theories and mainstream logics. At the moment, I only have new proofs of old results, and a vague intuition that some technical machinery is simpler. I think these are enough to show that LTTs have potential. They are a solution looking for a problem. I really need to get better at inventing names. Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 26 / 27

Conclusion LTTs might possibly be a useful tool for bridging type theories and mainstream logics. At the moment, I only have new proofs of old results, and a vague intuition that some technical machinery is simpler. I think these are enough to show that LTTs have potential. They are a solution looking for a problem. I really need to get better at inventing names. I hope someone in the audience is thinking LTTs might be the tool they ve been looking for. Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 26 / 27

Conclusion LTTs might possibly be a useful tool for bridging type theories and mainstream logics. At the moment, I only have new proofs of old results, and a vague intuition that some technical machinery is simpler. I think these are enough to show that LTTs have potential. They are a solution looking for a problem. I really need to get better at inventing names. I hope someone in the audience is thinking LTTs might be the tool they ve been looking for. Please come talk to me over lunch. Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 26 / 27

References Robin Adams and Zhaohui Luo. Classical predicative logic-enriched type theories. Annals of Pure and Applied Logic, 161:1315 1345, 2010. Robin Adams and Zhaohui Luo. Weyl s predicative classical mathematics as a logic-enriched type theory. ACM Transactions on Computational Logic., 11(2), 2010. Nicola Gambino and Peter Aczel. The generalised type-theoretic interpretation of constructive set theory. J. Symbolic Logic, 71(1):67 103, 2006. Robin Adams (RHUL) Ordinal Strength of LTTs 27 March 2012 27 / 27