Weak theories of truth and explicit mathematics

Transcription

1 Weak theories of truth and explicit mathematics Thomas Strahm (joint work with Sebastian Eberhard) Institut für Informatik und angewandte Mathematik, Universität Bern Oxford, Sep 19, 2011 T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

2 Introduction General aims of this talk In this talk we will discuss weak theories of truth over combinatory logic weak systems of Feferman s explicit mathematics the relationship between the two formalisms we consider two truth theories T PR and T PT of primitive recursive and polynomial time strength, respectively T PT is a novel abstract truth-theoretic framework which is able to interpret feasible subsystems of explicit mathematics the proof that T PT is feasible is non-trivial and due to Sebastian Eberhard (see his talk tomorrow) T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

3 Introduction Explicit mathematics Systems of explicit mathematics have been introduced by Feferman in They have been employed in foundational works in various ways: foundations of constructive mathematics proof theory of subsystems of second order arithmetic and set theory; foundational reductions logical foundations of functional and object-oriented programming languages universes and higher reflection principles formal proof-theoretic framework for abstract computations from ordinary and generalized recursion theory T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

4 Introduction Theories of truth over combinarory logic The truth theories over combinatory logic relevant to this talk have various roots: illative combinatory logic (Curry, Fitch) Frege structures (Scott, Aczel, Flagg and Myhill) Kripke-Feferman style axiomatizations of truth over combinatory algebras Intensively studied by Cantini (see e.g. his monograph Logical frameworks for truth and abstraction) and Kahle (see e.g. his Habilitationsschrift The applicative realm) T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

5 Outline 1 Introduction 2 The basic applicative framework 3 Adding types and names 4 Adding truth 5 Relating weak explicit mathematics and truth 6 Concluding remark T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

6 The basic applicative framework Informal applicative setting T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

7 The basic applicative framework Informal applicative setting Untyped universe of operations or rules, which can be freely applied to eachother T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

8 The basic applicative framework Informal applicative setting Untyped universe of operations or rules, which can be freely applied to eachother Self-application is meaningful, though not necessarily total T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

9 The basic applicative framework Informal applicative setting Untyped universe of operations or rules, which can be freely applied to eachother Self-application is meaningful, though not necessarily total The computational engine of these rules is given by a partial combinatory algebra, featuring partial versions of Curry s combinators k and s T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

10 The basic applicative framework Informal applicative setting Untyped universe of operations or rules, which can be freely applied to eachother Self-application is meaningful, though not necessarily total The computational engine of these rules is given by a partial combinatory algebra, featuring partial versions of Curry s combinators k and s In addition, there is a ground urelement structure of the binary words or strings with certain natural operations on them T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

11 The basic applicative framework Informal applicative setting (ctd.) Let W denote the set of (finite) binary words. We will consider the following operations: T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

12 The basic applicative framework Informal applicative setting (ctd.) Let W denote the set of (finite) binary words. We will consider the following operations: s 0 and s 1 : binary successors on W with predecessor p W T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

13 The basic applicative framework Informal applicative setting (ctd.) Let W denote the set of (finite) binary words. We will consider the following operations: s 0 and s 1 : binary successors on W with predecessor p W : word concatenation T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

14 The basic applicative framework Informal applicative setting (ctd.) Let W denote the set of (finite) binary words. We will consider the following operations: s 0 and s 1 : binary successors on W with predecessor p W : word concatenation : word multiplication T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

15 The basic applicative framework The logic of partial terms The logic of partial terms (LPT) due to Beeson/Feferman is a modification of first-order predicate logic taking into account partial functions. T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

16 The basic applicative framework The logic of partial terms The logic of partial terms (LPT) due to Beeson/Feferman is a modification of first-order predicate logic taking into account partial functions. Variables range over defined objects only T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

17 The basic applicative framework The logic of partial terms The logic of partial terms (LPT) due to Beeson/Feferman is a modification of first-order predicate logic taking into account partial functions. Variables range over defined objects only (Composed) terms do not necessarily denote and t signifies that t has a value T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

18 The basic applicative framework The logic of partial terms The logic of partial terms (LPT) due to Beeson/Feferman is a modification of first-order predicate logic taking into account partial functions. Variables range over defined objects only (Composed) terms do not necessarily denote and t signifies that t has a value The usual quantifier axioms of predicate logic are modified, e.g. we have A(t) t ( x)a(x) T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

19 The basic applicative framework The logic of partial terms The logic of partial terms (LPT) due to Beeson/Feferman is a modification of first-order predicate logic taking into account partial functions. Variables range over defined objects only (Composed) terms do not necessarily denote and t signifies that t has a value The usual quantifier axioms of predicate logic are modified, e.g. we have A(t) t ( x)a(x) Strictness axioms claim that terms occurring in positive atoms are defined T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

20 The basic applicative framework The basic applicative language L L is a first order language for the logic of partial terms: constants k, s, p, p 0, p 1, d W, ɛ, s 0, s 1, p W, c,,... relation symbols =,, W arbitrary term application T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

21 The basic applicative framework The basic applicative language L L is a first order language for the logic of partial terms: constants k, s, p, p 0, p 1, d W, ɛ, s 0, s 1, p W, c,,... relation symbols =,, W arbitrary term application Notation t 1 t 2... t n := (... (t 1 t 2 ) t n ) T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

22 The basic applicative framework The basic applicative language L L is a first order language for the logic of partial terms: constants k, s, p, p 0, p 1, d W, ɛ, s 0, s 1, p W, c,,... relation symbols =,, W arbitrary term application Notation t 1 t 2... t n := (... (t 1 t 2 ) t n ) t 1 t 2 := t 1 t 2 t 1 = t 2 T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

23 The basic applicative framework The basic applicative language L L is a first order language for the logic of partial terms: constants k, s, p, p 0, p 1, d W, ɛ, s 0, s 1, p W, c,,... relation symbols =,, W arbitrary term application Notation t 1 t 2... t n := (... (t 1 t 2 ) t n ) t 1 t 2 := t 1 t 2 t 1 = t 2 t W := W(t) T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

24 The basic applicative framework The basic applicative language L L is a first order language for the logic of partial terms: constants k, s, p, p 0, p 1, d W, ɛ, s 0, s 1, p W, c,,... relation symbols =,, W arbitrary term application Notation t 1 t 2... t n := (... (t 1 t 2 ) t n ) t 1 t 2 := t 1 t 2 t 1 = t 2 t W := W(t) t : W k W := ( x 1... x k W)tx 1... x k W T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

25 The basic applicative framework The basic applicative language L L is a first order language for the logic of partial terms: constants k, s, p, p 0, p 1, d W, ɛ, s 0, s 1, p W, c,,... relation symbols =,, W arbitrary term application Notation t 1 t 2... t n := (... (t 1 t 2 ) t n ) t 1 t 2 := t 1 t 2 t 1 = t 2 t W := W(t) t : W k W := ( x 1... x k W)tx 1... x k W s t := 1 s 1 t T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

26 The basic applicative framework The basic theory of operations and words B The logic of B is the logic of partial terms. The non-logical axioms of B include: partial combinatory algebra: kxy = x, sxy sxyz xz(yz) pairing p with projections p 0 and p 1 defining axioms for the binary words W with ɛ, the successors s 0, s 1 and the predecessor p W. definition by cases d W on W initial subword relation c word concatenation and word multiplication T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

27 The basic applicative framework Consequences of the partial combinatory algebra axioms As usual in untyped applicative settings we have: Lemma (Explicit definitions and fixed points) 1 For each L term t there exists an L term (λx.t) so that B (λx.t) (λx.t)x t 2 There is a closed L term fix so that B fixg fixgx g(fixg)x T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

28 The basic applicative framework Standard models Example (Recursion-theoretic model PRO) Take the universe of binary words and interpret application as partial recursive function application in the sense of o.r.t. T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

29 The basic applicative framework Standard models Example (Recursion-theoretic model PRO) Take the universe of binary words and interpret application as partial recursive function application in the sense of o.r.t. Example (The open term model M(λη)) Take the universe of open terms Consider the usual reduction of the extensional lambda calculus λη Application is juxtaposition Two terms are equal if they have a common reduct W denotes those terms that reduce to a standard word w Note that M(λη) satiesfies totality of application (Tot) and extensionality of operations (Ext) T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

30 The basic applicative framework Natural induction principles T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

31 The basic applicative framework Natural induction principles Σ b W -formulas Formulas A(x) of the form ( y W)(y fx B(f, x, y)) for B positive and W-free T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

32 The basic applicative framework Natural induction principles Σ b W -formulas Formulas A(x) of the form ( y W)(y fx B(f, x, y)) for B positive and W-free Σ b W notation induction on W, (Σb W -I W) f : W W A(ɛ) ( x W)(A(x) A(s 0 x) A(s 1 x)) ( x W)A(x) T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

33 The basic applicative framework Natural induction principles Σ b W -formulas Formulas A(x) of the form ( y W)(y fx B(f, x, y)) for B positive and W-free Σ b W notation induction on W, (Σb W -I W) f : W W A(ɛ) ( x W)(A(x) A(s 0 x) A(s 1 x)) ( x W)A(x) Positive induction on W, (Pos-I W ) Induction on W for arbitrary positive formulas. T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

34 The basic applicative framework Provably total functions Definition A function F : W n W is called provably total in an L theory T, if there exists a closed L term t F such that (i) T t F : W n W and, in addition, (ii) T t F w 1 w n = F (w 1,..., w n ) for all w 1,..., w n in W. Let τ (T) = {F : F provably total in T}. T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

35 The basic applicative framework Proof-theoretic results I The two systems PT and PR PT := B + (Σ b W -I W) PR := B + (Pos-I W ) T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

36 The basic applicative framework Proof-theoretic results I The two systems PT and PR PT := B + (Σ b W -I W) PR := B + (Pos-I W ) Theorem (Cantini) τ(pr) equals the class of primitive recursive functions. T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

37 The basic applicative framework Proof-theoretic results I The two systems PT and PR PT := B + (Σ b W -I W) PR := B + (Pos-I W ) Theorem (Cantini) τ(pr) equals the class of primitive recursive functions. Theorem (S.) τ(pt) equals the class of polynomial time computable functions. T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

38 Adding types and names 1 Introduction 2 The basic applicative framework 3 Adding types and names 4 Adding truth 5 Relating weak explicit mathematics and truth 6 Concluding remark T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

39 Adding types and names Types and names in explicit mathematics T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

40 Adding types and names Types and names in explicit mathematics Types are collections of individuals and can have quite complicated defining properties T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

41 Adding types and names Types and names in explicit mathematics Types are collections of individuals and can have quite complicated defining properties Types are represented by operations or names T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

42 Adding types and names Types and names in explicit mathematics Types are collections of individuals and can have quite complicated defining properties Types are represented by operations or names Each type may have several different names or representations T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

43 Adding types and names Types and names in explicit mathematics Types are collections of individuals and can have quite complicated defining properties Types are represented by operations or names Each type may have several different names or representations The interplay of names and types on the level of operations witnesses the explicit character of explicit mathematics T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

44 Adding types and names Types and names in explicit mathematics Types are collections of individuals and can have quite complicated defining properties Types are represented by operations or names Each type may have several different names or representations The interplay of names and types on the level of operations witnesses the explicit character of explicit mathematics In the following we use a formalization of the types and names paradigm due to Jäger T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

45 Adding types and names The language of types and names The language L is a two-sorted language extending L by T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

46 Adding types and names The language of types and names The language L is a two-sorted language extending L by type variables U, V, W, X, Y, Z,... binary relation symbols R (naming) and (elementhood) new (individual) constants w (binary words W), id (identity), dom (domain), un (union), int (intersection), inv (inverse image), and all (universal quantification) T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

47 Adding types and names The language of types and names The language L is a two-sorted language extending L by type variables U, V, W, X, Y, Z,... binary relation symbols R (naming) and (elementhood) new (individual) constants w (binary words W), id (identity), dom (domain), un (union), int (intersection), inv (inverse image), and all (universal quantification) The formulas A, B, C,... of L are built from the atomic formulas of L as well as from formulas of the form (s X ), R(s, X ), (X = Y ) by closing under the boolean connectives and quantification in both sorts. T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

48 Adding types and names Ontological axioms We use the following abbreviations: R(s) := X R(s, X ), s t := X (R(t, X ) s X ). T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

49 Adding types and names Ontological axioms We use the following abbreviations: R(s) := X R(s, X ), s t := X (R(t, X ) s X ). Ontological axioms (explicit representation and extensionality) (O1) (O2) (O3) xr(x, X ) R(a, X ) R(a, Y ) X = Y z(z X z Y ) X = Y T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

50 Adding types and names The system EPC Type existence axioms (w) (id) (inv) (un) (int) (dm) (all) R(w) x(x w W(x)) R(id) x(x id y(x = (y, y))) R(a) R(inv(f, a)) x(x inv(f, a) fx a) R(a) R(b) R(un(a, b)) x(x un(a, b) (x a x b)) R(a) R(b) R(int(a, b)) x(x int(a, b) (x a x b)) R(a) R(dom(a)) x(x dom(a) y((x, y) a)) R(a) R(all(a)) x(x all(a) y((x, y) a)) T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

51 Adding types and names The system EPC (continued) Type induction on W ɛ X ( x W)(x X s 0 x X s 1 x X ) ( x W)(x X ) Definition (The theory EPC) EPC is the extension of the first-order applicative theory B by the ontological axioms the above type existence axioms type induction on W T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

52 Adding types and names The system PET PET is a subsystem of EPC where it is no longer claimed that the class W of words forms a type. T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

53 Adding types and names The system PET PET is a subsystem of EPC where it is no longer claimed that the class W of words forms a type. Definition Define W a (x) := W(x) x a. T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

54 Adding types and names The system PET PET is a subsystem of EPC where it is no longer claimed that the class W of words forms a type. Definition Define W a (x) := W(x) x a. Only initial segments of W define types: (w a ) a W R(w(a)) x(x w(a) W a (x)) All other axioms of PET are identical to the ones of EPC. T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

55 Adding types and names The Join axiom The Join axioms are given by the following assertions (J.1) and (J.2): (J.1) (J.2) R(a) ( x a)r(fx) R(j(a, f )) R(a) ( x a)r(fx) x(x j(a, f ) Σ[f, a, x]) where Σ[f, a, x] is the formula y z(x = (y, z) y a z fy) Let us write EPCJ and PETJ for the extension of EPC and PET by the join principle. T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

56 Adding types and names Proof-theoretic results II Theorem (Cantini) τ(epcj) equals the class of primitive recursive functions. T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

57 Adding types and names Proof-theoretic results II Theorem (Cantini) τ(epcj) equals the class of primitive recursive functions. Theorem (Spescha, S.) τ(petj i ) equals the class of polynomial time computable functions. T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

58 Adding types and names Proof-theoretic results II Theorem (Cantini) τ(epcj) equals the class of primitive recursive functions. Theorem (Spescha, S.) τ(petj i ) equals the class of polynomial time computable functions. Theorem (Probst) τ(petj) equals the class of polynomial time computable functions. T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

59 Adding truth 1 Introduction 2 The basic applicative framework 3 Adding types and names 4 Adding truth 5 Relating weak explicit mathematics and truth 6 Concluding remark T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

60 Adding truth The language L T of positive truth The (first order) language L T is an extension of the language L by T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

61 Adding truth The language L T of positive truth The (first order) language L T is an extension of the language L by a new unary predicate symbol T for truth new individual constants =, Ẇ,,,, T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

62 Adding truth The language L T of positive truth The (first order) language L T is an extension of the language L by a new unary predicate symbol T for truth new individual constants =, Ẇ,,,, The new constants allow only the coding of positive formulas since we do not add a constant for negation. As usual, we will use infix notation for =,, and. T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

63 Adding truth The language L T of positive truth The (first order) language L T is an extension of the language L by a new unary predicate symbol T for truth new individual constants =, Ẇ,,,, The new constants allow only the coding of positive formulas since we do not add a constant for negation. As usual, we will use infix notation for =,, and. The formulas of L T are built as expected, where for each term t, T(t) is a new atomic formula. T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

64 Adding truth The truth theory T PR Positive truth axioms T(x =y) x = y T(Ẇx) W(x) T(x y) T(x) T(y) T(x y) T(x) T(y) T( f ) xt(fx) T( f ) xt(fx) T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

65 Adding truth The truth theory T PR (continued) Truth induction on W T(rɛ) ( x W)(T(rx) T(r(s 0 x)) T(r(s 1 x))) ( x W)T(rx) Definition (The theory T PR ) T PR is the extension of the first-order applicative theory B by totality of application the above truth axioms truth induction on W T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

66 Adding truth The truth theory T PT T PT is a subsystem of T PR where the truth predicate can only reflect initial segments of the predicate W. T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

67 Adding truth The truth theory T PT T PT is a subsystem of T PR where the truth predicate can only reflect initial segments of the predicate W. Recall that W a (x) := W(x) x a. T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

68 Adding truth The truth theory T PT T PT is a subsystem of T PR where the truth predicate can only reflect initial segments of the predicate W. Recall that W a (x) := W(x) x a. Only initial segments of W are reflected by T: a W (T(Ẇax) W a (x)) All other axioms of T PT are identical to the ones of T PR. T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

69 Adding truth Proof-theoretic results III Theorem (Cantini) τ(t PR ) equals the class of primitive recursive functions. T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

70 Adding truth Proof-theoretic results III Theorem (Cantini) τ(t PR ) equals the class of primitive recursive functions. Theorem (Eberhard) τ(t PT ) equals the class of polynomial time computable functions. T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

71 Relating weak explicit mathematics and truth 1 Introduction 2 The basic applicative framework 3 Adding types and names 4 Adding truth 5 Relating weak explicit mathematics and truth 6 Concluding remark T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

72 Relating weak explicit mathematics and truth Embedding explicit mathematics into truth theories T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

73 Relating weak explicit mathematics and truth Embedding explicit mathematics into truth theories The translation of EPCJ and PETJ into T PR and T PT, respectively, works very easily. Basically, define in such a way that (s t) is simply T(t s ) T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

74 Relating weak explicit mathematics and truth Embedding explicit mathematics into truth theories The translation of EPCJ and PETJ into T PR and T PT, respectively, works very easily. Basically, define in such a way that (s t) is simply T(t s ) The type constructors are interpreted by terms of L T which embody their membership conditions T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

75 Relating weak explicit mathematics and truth Embedding explicit mathematics into truth theories The translation of EPCJ and PETJ into T PR and T PT, respectively, works very easily. Basically, define in such a way that (s t) is simply T(t s ) The type constructors are interpreted by terms of L T which embody their membership conditions Here we implicitly assume a first-order formulation of explicit mathematics T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

76 Relating weak explicit mathematics and truth Embedding explicit mathematics into truth theories (ctd.) Translation of the type constructors in the case of EPCJ id λz. λy. z = y, y w λz. Ẇz int λa.λb.λz. az bz un λa.λb.λz. az bz inv λf.λa.λz. a(fz) dom λa.λz. λy.a z, y all λa.λz. λy.a z, y j λf.λa.λz. λx. λy.z = x, y ax (fx)y T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

77 Relating weak explicit mathematics and truth Embedding truth theories into explicit mathematics T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

78 Relating weak explicit mathematics and truth Embedding truth theories into explicit mathematics The direct embedding of weak truth theories into explicit mathematics requires additional assumptions T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

79 Relating weak explicit mathematics and truth Embedding truth theories into explicit mathematics The direct embedding of weak truth theories into explicit mathematics requires additional assumptions Namely, we employ the existence of universes and Cantini s uniformity principle T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

80 Relating weak explicit mathematics and truth Embedding truth theories into explicit mathematics The direct embedding of weak truth theories into explicit mathematics requires additional assumptions Namely, we employ the existence of universes and Cantini s uniformity principle These principles do not raise the proof-theoretic strength of the underlying formalisms T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

81 Relating weak explicit mathematics and truth Cantini s uniformity principle Uniformity principle (Cantini) (UP) x( y W)A[x, y] ( y W) xa[x, y] where A[x, y] is positive elementary. T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

82 Relating weak explicit mathematics and truth Universes in explicit mathematics T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

83 Relating weak explicit mathematics and truth Universes in explicit mathematics A universe in explicit mathematics is a type of names which is closed under previously recognized type existence principles T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

84 Relating weak explicit mathematics and truth Universes in explicit mathematics A universe in explicit mathematics is a type of names which is closed under previously recognized type existence principles An EPCJ universe is closed under the type constructors of the theory EPCJ; analogously for PETJ universes T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

85 Relating weak explicit mathematics and truth Universes in explicit mathematics A universe in explicit mathematics is a type of names which is closed under previously recognized type existence principles An EPCJ universe is closed under the type constructors of the theory EPCJ; analogously for PETJ universes EPCJ + U is the extension of EPCJ where it is claimed that each type is contained in an EPCJ universe; analogously for PETJ + U T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

86 Relating weak explicit mathematics and truth Reducing T PR to EPCJ + U + UP T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

87 Relating weak explicit mathematics and truth Reducing T PR to EPCJ + U + UP the main task is to define a type which interprets the truth predicate T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

88 Relating weak explicit mathematics and truth Reducing T PR to EPCJ + U + UP the main task is to define a type which interprets the truth predicate ω many levels suffice; we work in a universe in order to show that the truth level type τw is indeed a name for each w W T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

89 Relating weak explicit mathematics and truth Reducing T PR to EPCJ + U + UP the main task is to define a type which interprets the truth predicate ω many levels suffice; we work in a universe in order to show that the truth level type τw is indeed a name for each w W UP is used in order to deal with the axioms about T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

90 Relating weak explicit mathematics and truth Reducing T PT to PETJ + U T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

91 Relating weak explicit mathematics and truth Reducing T PT to PETJ + U the reduction is more delicate since we cannot collect the truth levels for all words into a type T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

92 Relating weak explicit mathematics and truth Reducing T PT to PETJ + U the reduction is more delicate since we cannot collect the truth levels for all words into a type there is an intermediate step via an asymmetric interpretation to a leveled truth theory T l PT where the levels of the truth predicate are polynomially bounded in a specific way T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

93 Relating weak explicit mathematics and truth Reducing T PT to PETJ + U the reduction is more delicate since we cannot collect the truth levels for all words into a type there is an intermediate step via an asymmetric interpretation to a leveled truth theory T l PT where the levels of the truth predicate are polynomially bounded in a specific way further T l PT can be modeled in PETJ + U similarly as above T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

94 Relating weak explicit mathematics and truth Reducing T PT to PETJ + U the reduction is more delicate since we cannot collect the truth levels for all words into a type there is an intermediate step via an asymmetric interpretation to a leveled truth theory T l PT where the levels of the truth predicate are polynomially bounded in a specific way further T l PT can be modeled in PETJ + U similarly as above it should be noted that the direct proof-theoretic treatment of PETJ + U requires Eberhard s involved new realizability techniques developed for T PT T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

95 Concluding remark 1 Introduction 2 The basic applicative framework 3 Adding types and names 4 Adding truth 5 Relating weak explicit mathematics and truth 6 Concluding remark T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

96 Concluding remark Applications of T PT to the unfolding program T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

97 Concluding remark Applications of T PT to the unfolding program the feasible truth theory T PT is also an important reference theory for our recent work on Feferman s unfolding program T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

98 Concluding remark Applications of T PT to the unfolding program the feasible truth theory T PT is also an important reference theory for our recent work on Feferman s unfolding program we are working on the unfolding of a natural schematic system FEA of feasible arithmetic T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

99 Concluding remark Applications of T PT to the unfolding program the feasible truth theory T PT is also an important reference theory for our recent work on Feferman s unfolding program we are working on the unfolding of a natural schematic system FEA of feasible arithmetic the system T PT plays a crucial role in order to obtain proof-theoretic upper bounds for the full predicate unfolding U(FEA) of FEA T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41

100 Concluding remark Selected References 1 A. Cantini, Proof-theoretical aspects of self-referential truth, LMPS 95, Kluwer, A. Cantini, Choice and uniformity in weak applicative theories, Logic Colloquium 01, LNL 20, ASL, S. Eberhard, A truth theory over an applicative framework of strength PT, in preparation 4 S. Eberhard, T. Strahm, Weak theories of truth and explicit mathematics, submitted. 5 D. Probst, The provably terminating operations of the subsystem PETJ of explicit mathematics, Annals of Pure and Applied Logic 162, D. Spescha, T. Strahm, Elementary explicit types and polynomial time operations, Mathematical Logic Quarterly 55, D. Spescha, T. Strahm, Realisability in weak systems of explicit mathematics, Mathematical Logic Quarterly (to appear) 8 T. Strahm, Theories with self-application and computational complexity, Information and Computation 185, 2003 T. Strahm (IAM, Univ. Bern) Weak theories of truth Oxford, Sep 19, / 41