## Weak theories of truth and explicit mathematics. Submitted for publication. 19

Citations: | 2 - 2 self |

### BibTeX

@MISC{Eberhard_weaktheories,

author = {Sebastian Eberhard and Thomas Strahm},

title = {Weak theories of truth and explicit mathematics. Submitted for publication. 19},

year = {}

}

### OpenURL

### Abstract

We study weak theories of truth over combinatory logic and their relationship to weak systems of explicit mathematics. In particular, we consider two truth theories TPR and TPT of primitive recursive and feasible strength. The latter theory is a novel abstract truth-theoretic setting which is able to interpret expressive feasible subsystems of explicit mathematics. 1

### Citations

90 |
A language and axioms for explicit mathematics
- Feferman
- 1975
(Show Context)
Citation Context ...ng principles of recursion are available due to the known expressivity of combinatory algebras. The first order applicative base describes the operational core of Feferman’s explicit mathematics, cf. =-=[15, 16, 17]-=-; instead of a predicate N for natural numbers we will consider a predicate W in order to single out those operations which denote binary words. ∗Institut für Informatik und angewandte Mathematik, Uni... |

72 |
Foundations of constructive mathematics. Metamathematical studies. Ergebnisse der Mathematik und ihrer Grenzgebiete (3
- Beeson
- 1985
(Show Context)
Citation Context ...(ax). The so-obtained axiomatic frameworks of partial, self-referential truth are rooted in Kripke’s seminal work and also yield an interpretation of classical Frege structures (cf. Aczel [1], Beeson =-=[2]-=-, and Hayashi and Kobayashi [27]). For detailed background on the type of truth theories considered here, see Cantini [6, 7] and Kahle [30, 31]. Of course, the work on axiomatic truth over combinatory... |

66 |
Constructive theories of functions and classes
- Feferman
- 1979
(Show Context)
Citation Context ...ng principles of recursion are available due to the known expressivity of combinatory algebras. The first order applicative base describes the operational core of Feferman’s explicit mathematics, cf. =-=[15, 16, 17]-=-; instead of a predicate N for natural numbers we will consider a predicate W in order to single out those operations which denote binary words. ∗Institut für Informatik und angewandte Mathematik, Uni... |

61 |
Frege structures and the notions of proposition, truth and set. The Kleene Symposium
- Aczel
- 1980
(Show Context)
Citation Context ...g x ∈ a as T(ax). The so-obtained axiomatic frameworks of partial, self-referential truth are rooted in Kripke’s seminal work and also yield an interpretation of classical Frege structures (cf. Aczel =-=[1]-=-, Beeson [2], and Hayashi and Kobayashi [27]). For detailed background on the type of truth theories considered here, see Cantini [6, 7] and Kahle [30, 31]. Of course, the work on axiomatic truth over... |

48 | Systems of explicit mathematics with non-constructive µ-operator
- Feferman, Jäger
- 1993
(Show Context)
Citation Context ...am to use explicit mathematics to analyze properties of functional programs. In the realm of pure applicative theories, natural systems of strength PRA are proposed and analyzed in Feferman and Jäger =-=[22]-=- and Jäger and Strahm [29]. Theories that are strongly related to the system EPCJ are considered in Krähenbühl [33]. Let us note that all upper bound computations for the systems mentioned above proce... |

23 |
Proving programs and programming proofs
- Beeson
- 1986
(Show Context)
Citation Context ...binary word w ∈ W. 2.2 The basic theory of operations and words B The applicative base theory B has been introduced in Strahm [40, 41]. Its logic is the classical logic of partial terms due to Beeson =-=[2, 3]-=-. The nonlogical axioms of B include: 4• partial combinatory algebra: kxy = x, sxy↓ ∧ sxyz ≃ xz(yz) • pairing p with projections p0 and p1 • defining axioms for the binary words W with ɛ, the binary ... |

23 |
Logics for termination and correctness of functional programs II: Logics of strength PRA
- Feferman
- 1992
(Show Context)
Citation Context ...i [6, 7] and Kahle [30, 31]. Of course, the work on axiomatic truth over combinatory logic is also strongly related to corresponding work in the area of arithmetical truth theories, see e.g. Feferman =-=[18, 21]-=-, Friedman and Sheard [25], and Halbach [26]. The focus of the present paper is to discuss various weak (positive) truth theories and systems of explicit mathematics as well as their mutual relationsh... |

23 |
An axiomatic approach to selfreferential truth
- Friedman, Sheard
- 1987
(Show Context)
Citation Context ...f course, the work on axiomatic truth over combinatory logic is also strongly related to corresponding work in the area of arithmetical truth theories, see e.g. Feferman [18, 21], Friedman and Sheard =-=[25]-=-, and Halbach [26]. The focus of the present paper is to discuss various weak (positive) truth theories and systems of explicit mathematics as well as their mutual relationship. Namely we will address... |

21 | On applicative theories
- Jäger, Kahle, et al.
- 1999
(Show Context)
Citation Context ...tics to analyze properties of functional programs. In the realm of pure applicative theories, natural systems of strength PRA are proposed and analyzed in Feferman and Jäger [22] and Jäger and Strahm =-=[29]-=-. Theories that are strongly related to the system EPCJ are considered in Krähenbühl [33]. Let us note that all upper bound computations for the systems mentioned above proceed via embeddings into sui... |

16 | Gödel’s program for new axioms: Why, where, how and what
- Feferman
- 1996
(Show Context)
Citation Context ...proof-theoretic strength of the underlying systems. The newly proposed feasible truth theory TPT is also an important reference theory in our recent work on Feferman’s unfolding program (see Feferman =-=[20]-=- and Feferman and Strahm [23, 24]): in Eberhard and Strahm [13, 14], the system TPT plays a crucial role in order to obtain proof-theoretic upper bounds for the full unfolding U(FEA) of a natural sche... |

15 |
Induction in the elementary theory of types and names
- Jäger
- 1988
(Show Context)
Citation Context ...operations via a suitable naming relation ℜ. Types are extensional and have (explicit) names which are intensional. The formalization of explicit mathematics using a naming relation ℜ is due to Jäger =-=[28]-=-. We will present the two weak theories of explicit mathematics EPCJ and PETJ and some extensions thereof. We will describe the two theories simultaneously since their axioms differ only slightly. 4.1... |

14 |
Logical Frameworks for Truth and Abstraction
- Cantini
- 1996
(Show Context)
Citation Context ...o yield an interpretation of classical Frege structures (cf. Aczel [1], Beeson [2], and Hayashi and Kobayashi [27]). For detailed background on the type of truth theories considered here, see Cantini =-=[6, 7]-=- and Kahle [30, 31]. Of course, the work on axiomatic truth over combinatory logic is also strongly related to corresponding work in the area of arithmetical truth theories, see e.g. Feferman [18, 21]... |

12 |
combinatory logic and positive safe induction
- Cantini
(Show Context)
Citation Context ...gth PRA. Let us now turn to the discussion of the proof theory of systems of explicit mathematics of polynomial strength. Purely first order systems were proposed and analyzed in Calamai [4], Cantini =-=[8, 9, 10, 5]-=-, Kahle and Oitavem [32], and Strahm [39, 40, 41]. 5 For those theories, a direct embedding into feasible subsystems of arithmetic does not seem to be possible, as, for example, already the standard i... |

12 | Theories with self-application and computational complexity
- Strahm
(Show Context)
Citation Context ...n A. Finally, let us write w for the canonical closed L term denoting the binary word w ∈ W. 2.2 The basic theory of operations and words B The applicative base theory B has been introduced in Strahm =-=[40, 41]-=-. Its logic is the classical logic of partial terms due to Beeson [2, 3]. The nonlogical axioms of B include: 4• partial combinatory algebra: kxy = x, sxy↓ ∧ sxyz ≃ xz(yz) • pairing p with projection... |

11 | and uniformity in weak applicative theories
- Cantini
(Show Context)
Citation Context ...cative framework which is common to all systems studied in this paper. Section 3 presents the two central truth theories of this paper, TPR and TPT. The first one was previously introduced in Cantini =-=[10, 7]-=-. Both systems rely on a form of positive truth and embody truth induction. Whereas TPR can reflect the whole predicate W of binary words, TPT only reflects initial segments. In Section 4 we will pres... |

9 |
Proof-theoretic aspects of self-referential truth
- Cantini
- 1995
(Show Context)
Citation Context ...o yield an interpretation of classical Frege structures (cf. Aczel [1], Beeson [2], and Hayashi and Kobayashi [27]). For detailed background on the type of truth theories considered here, see Cantini =-=[6, 7]-=- and Kahle [30, 31]. Of course, the work on axiomatic truth over combinatory logic is also strongly related to corresponding work in the area of arithmetical truth theories, see e.g. Feferman [18, 21]... |

8 |
Applikative Theorien und Frege-Strukturen. PhD thesis, Institut für Informatik und angewandte Mathematik
- Kahle
- 1997
(Show Context)
Citation Context ...retation of classical Frege structures (cf. Aczel [1], Beeson [2], and Hayashi and Kobayashi [27]). For detailed background on the type of truth theories considered here, see Cantini [6, 7] and Kahle =-=[30, 31]-=-. Of course, the work on axiomatic truth over combinatory logic is also strongly related to corresponding work in the area of arithmetical truth theories, see e.g. Feferman [18, 21], Friedman and Shea... |

7 | Polynomial time operations in explicit mathematics
- Strahm
- 1997
(Show Context)
Citation Context ...proof theory of systems of explicit mathematics of polynomial strength. Purely first order systems were proposed and analyzed in Calamai [4], Cantini [8, 9, 10, 5], Kahle and Oitavem [32], and Strahm =-=[39, 40, 41]-=-. 5 For those theories, a direct embedding into feasible subsystems of arithmetic does not seem to be possible, as, for example, already the standard interpretation of equality of terms translates int... |

6 |
Feasible operations and applicative theories based on λη
- Cantini
(Show Context)
Citation Context ...gth PRA. Let us now turn to the discussion of the proof theory of systems of explicit mathematics of polynomial strength. Purely first order systems were proposed and analyzed in Calamai [4], Cantini =-=[8, 9, 10, 5]-=-, Kahle and Oitavem [32], and Strahm [39, 40, 41]. 5 For those theories, a direct embedding into feasible subsystems of arithmetic does not seem to be possible, as, for example, already the standard i... |

6 | Weak systems of explicit mathematics
- Spescha
- 2009
(Show Context)
Citation Context ...reflects initial segments. In Section 4 we will present two natural systems of explicit mathematics of polynomial and primitive recursive strength, respectively: the system PETJ of Spescha and Strahm =-=[37, 36, 38]-=- and a natural explicit system EPCJ; both of these frameworks are direct subsystems of Feferman’s EM0 plus the join principle (cf. [15, 17]). For the embedding of truth theories into explicit mathemat... |

6 | Elementary explicit types and polynomial time operations
- Spescha, Strahm
(Show Context)
Citation Context ...reflects initial segments. In Section 4 we will present two natural systems of explicit mathematics of polynomial and primitive recursive strength, respectively: the system PETJ of Spescha and Strahm =-=[37, 36, 38]-=- and a natural explicit system EPCJ; both of these frameworks are direct subsystems of Feferman’s EM0 plus the join principle (cf. [15, 17]). For the embedding of truth theories into explicit mathemat... |

5 | The unfolding of non-finitist arithmetic
- Feferman, Strahm
(Show Context)
Citation Context ...he underlying systems. The newly proposed feasible truth theory TPT is also an important reference theory in our recent work on Feferman’s unfolding program (see Feferman [20] and Feferman and Strahm =-=[23, 24]-=-): in Eberhard and Strahm [13, 14], the system TPT plays a crucial role in order to obtain proof-theoretic upper bounds for the full unfolding U(FEA) of a natural schematic system FEA of feasible arit... |

5 | Proof-theoretic Contributions to Explicit Mathematics
- Strahm
- 2001
(Show Context)
Citation Context ...n A. Finally, let us write w for the canonical closed L term denoting the binary word w ∈ W. 2.2 The basic theory of operations and words B The applicative base theory B has been introduced in Strahm =-=[40, 41]-=-. Its logic is the classical logic of partial terms due to Beeson [2, 3]. The nonlogical axioms of B include: 4• partial combinatory algebra: kxy = x, sxy↓ ∧ sxyz ≃ xz(yz) • pairing p with projection... |

4 |
Remarks on applicative theories. Annals of Pure and Applied Logic 136
- Cantini
- 2005
(Show Context)
Citation Context ... counterparts using a forcing relation. The computational content of the intuitionistic truth theories is analyzed by means of suitable realizability techniques. A direct companion to [10] is Cantini =-=[11]-=-, which deals with further extensions of the theories in [10]. Let us note that our truth theory TPR can be embedded into PRA plus Σ 0 1 induction by using the formalized term model construction used ... |

4 |
The Applicative Realm. Habilitation Thesis
- Kahle
- 2007
(Show Context)
Citation Context ...retation of classical Frege structures (cf. Aczel [1], Beeson [2], and Hayashi and Kobayashi [27]). For detailed background on the type of truth theories considered here, see Cantini [6, 7] and Kahle =-=[30, 31]-=-. Of course, the work on axiomatic truth over combinatory logic is also strongly related to corresponding work in the area of arithmetical truth theories, see e.g. Feferman [18, 21], Friedman and Shea... |

4 | The provably terminating operations of the subsystem PETJ of explicit mathematics
- Probst
(Show Context)
Citation Context ...s for which the Tarski truth biconditionals hold. In the case of TPT this is the class of so-called simple formulas, which are patterned after similar classes of formulas in explicit mathematics, see =-=[37, 34]-=- and the next section of this paper. Definition 5 (Simple formulas) Let A be a positive LT formula and u be a variable not occurring in A. Then the formula Au which is obtained by replacing each subfo... |

4 | Weak theories of operations and types
- Strahm
(Show Context)
Citation Context ...elicate and difficult problem of showing that the provably total operations of the system PET with the join axiom and classical logic are still the polynomial time computable ones. The article Strahm =-=[42]-=- surveys most of these results. However, let use mention that the techniques used in these papers do not validate expansions of PETJ by universes or by the assertion ∀ℜ claiming that everything is a n... |

3 |
Proof-theoretic contributions to computational complexity
- Calamai
- 2008
(Show Context)
Citation Context ...etic of strength PRA. Let us now turn to the discussion of the proof theory of systems of explicit mathematics of polynomial strength. Purely first order systems were proposed and analyzed in Calamai =-=[4]-=-, Cantini [8, 9, 10, 5], Kahle and Oitavem [32], and Strahm [39, 40, 41]. 5 For those theories, a direct embedding into feasible subsystems of arithmetic does not seem to be possible, as, for example,... |

3 | Unfolding finitist arithmetic
- Feferman, Strahm
(Show Context)
Citation Context ...he underlying systems. The newly proposed feasible truth theory TPT is also an important reference theory in our recent work on Feferman’s unfolding program (see Feferman [20] and Feferman and Strahm =-=[23, 24]-=-): in Eberhard and Strahm [13, 14], the system TPT plays a crucial role in order to obtain proof-theoretic upper bounds for the full unfolding U(FEA) of a natural schematic system FEA of feasible arit... |

3 |
Explicit mathematics with positive existential comprehension and join
- Krähenbühl
- 2006
(Show Context)
Citation Context ...ies, natural systems of strength PRA are proposed and analyzed in Feferman and Jäger [22] and Jäger and Strahm [29]. Theories that are strongly related to the system EPCJ are considered in Krähenbühl =-=[33]-=-. Let us note that all upper bound computations for the systems mentioned above proceed via embeddings into suitable subsystems of Peano arithmetic of strength PRA. Let us now turn to the discussion o... |

2 |
A truth theory over an applicative framework of strength PT. Preliminary draft
- Eberhard
- 2011
(Show Context)
Citation Context ...sumptions. A further novelty of this paper is the introduction of a natural feasible truth theory TPT, whose provably total operations are the polynomial time computable ones, as is shown in Eberhard =-=[12]-=-. TPT can only reflect initial segments of the class W of binary words, but features unrestricted truth induction; it is obtained as a natural restriction of a truth theory TPR of the strength of prim... |

2 | Towards the unfolding of feasible arithmetic (Abstract - Eberhard, Strahm |

2 |
Axioms for the determinateness of truth
- Feferman
(Show Context)
Citation Context ...i [6, 7] and Kahle [30, 31]. Of course, the work on axiomatic truth over combinatory logic is also strongly related to corresponding work in the area of arithmetical truth theories, see e.g. Feferman =-=[18, 21]-=-, Friedman and Sheard [25], and Halbach [26]. The focus of the present paper is to discuss various weak (positive) truth theories and systems of explicit mathematics as well as their mutual relationsh... |

2 |
An applicative theory for FPH
- Kahle, Oitavem
(Show Context)
Citation Context ...iscussion of the proof theory of systems of explicit mathematics of polynomial strength. Purely first order systems were proposed and analyzed in Calamai [4], Cantini [8, 9, 10, 5], Kahle and Oitavem =-=[32]-=-, and Strahm [39, 40, 41]. 5 For those theories, a direct embedding into feasible subsystems of arithmetic does not seem to be possible, as, for example, already the standard interpretation of equalit... |

1 | Unfolding feasible arithmetic - Eberhard, Strahm |

1 |
A new fomalization of Feferman’s system of functions and classes and its relation to Frege structurs
- Hayashi, Kobayashi
- 1995
(Show Context)
Citation Context ... frameworks of partial, self-referential truth are rooted in Kripke’s seminal work and also yield an interpretation of classical Frege structures (cf. Aczel [1], Beeson [2], and Hayashi and Kobayashi =-=[27]-=-). For detailed background on the type of truth theories considered here, see Cantini [6, 7] and Kahle [30, 31]. Of course, the work on axiomatic truth over combinatory logic is also strongly related ... |