Unfolding Feasible Arithmetic andWeak Truth

Chapter
Part of the Logic, Epistemology, and the Unity of Science book series (LEUS, volume 36)

Abstract

In this paper we continue Feferman’s unfolding program initiated in (Feferman, vol. 6 of Lecture Notes in Logic, 1996) which uses the concept of the unfolding \(\mathcal{U}(\textsf{S})\) of a schematic system \(\textsf{S}\) in order to describe those operations, predicates and principles concerning them, which are implicit in the acceptance of \(\textsf{S}\). The program has been carried through for a schematic system of non-finitist arithmetic \(\textsf{NFA}\) in Feferman and Strahm (Ann Pure Appl Log, 104(1–3):75–96, 2000) and for a system \(\textsf{FA}\) (with and without Bar rule) in Feferman and Strahm (Rev Symb Log, 3(4):665–689, 2010). The present contribution elucidates the concept of unfolding for a basic schematic system \({\mathsf{FEA}}\) of feasible arithmetic. Apart from the operational unfolding \({\mathcal{U}_0(\textsf{FEA})}\) of \({\mathsf{FEA}}\), we study two full unfolding notions, namely the predicate unfolding \({\mathcal{U}(\textsf{FEA})}\) and a more general truth unfolding \({\mathcal{U}_{\mathsf{T}}(\textsf{FEA})}\) of \({\mathsf{FEA}}\), the latter making use of a truth predicate added to the language of the operational unfolding. The main results obtained are that the provably convergent functions on binary words for all three unfolding systems are precisely those being computable in polynomial time. The upper bound computations make essential use of a specific theory of truth \(\mathsf{T_{PT}}\) over combinatory logic, which has recently been introduced in Eberhard and Strahm (Bull Symb Log, 18(3):474–475, 2012) and Eberhard (A feasible theory of truth over combinatory logic, 2014) and whose involved proof-theoretic analysis is due to Eberhard (A feasible theory of truth over combinatory logic, 2014). The results of this paper were first announced in (Eberhard and Strahm, Bull Symb Log 18(3):474–475, 2012).

Keywords

Operational unfolding Predicate unfolding Truth unfolding Schematic system Feasible arithmetic 

References

  1. Cantini, A. (1996). Logical frameworks for truth and abstraction. Amsterdam: North-Holland.Google Scholar
  2. Cantini, A. (1997). Proof-theoretic aspects of self-referential truth. In M. L. D. Chiara, et al. (Ed.), Tenth international congress of logic, methodology and philosophy of science, Florence, August 1995 (Vol. 1, pp. 7–27). Dodrecht: Kluwer.Google Scholar
  3. Cantini, A. (2005). Choice and uniformity in weak applicative theories. In M. Baaz, S. Friedman, & J. Krajíček (Eds.), Logic colloquium ’01, vol. 20 of lecture notes in logic (pp. 108–138).Association for Symbolic Logic: A K Peters, Wellesley.Google Scholar
  4. Clote, P. (1999). Computation models and function algebras. In E. Griffor (Ed.) Handbook of computability theory (pp. 589–681) Amsterdam: Elsevier.Google Scholar
  5. Cobham, A. (1965). The intrinsic computational difficulty of functions. In Logic, methodology and philosophy of science II (pp. 24–30). Amsterdam: North Holland.Google Scholar
  6. Eberhard, S. A. (2014). Feasible theory of truth over combinatory logic. Annals of Pure and Applied Logic, 165(5), 1009–1033.Google Scholar
  7. Eberhard, S., & Strahm, T. (2012a). Towards the unfolding of feasible arithmetic (Abstract). Bulletin of Symbolic Logic, 18(3), 474–475.Google Scholar
  8. Eberhard, S., & Strahm, T. (2012b). Weak theories of truth and explicit mathematics. In U. Berger, H. Diener, P. Schuster, & M. Seisenberger (Eds.), Logic, construction, computation (pp. 157–184). Ontos.Google Scholar
  9. Feferman, S. (1975). A language and axioms for explicit mathematics. In J. Crossley (Ed.), Algebra and Logic, vol. 450 of lecture notes in mathematics (pp. 87–139). Berlin: Springer.Google Scholar
  10. Feferman, S. (1979). Constructive theories of functions and classes. In M. Boffa, D. van Dalen, & K. McAloon (Eds.), Logic colloquium '78 (pp. 159–224). Amsterdam: North Holland.Google Scholar
  11. Feferman, S. (1991). Reflecting on incompleteness. Journal of symbolic logic, 56(1), 1–49.CrossRefGoogle Scholar
  12. Feferman, S. (1996). Gödel’s program for new axioms: Why, where, how and what? In P. Hájek (Ed.), Gödel '96 vol. 6 of Lecture Notes in Logic (pp. 3–22). Berlin: Springer.Google Scholar
  13. Feferman, S. (2005). Predicativity. In S. Shapiro (Ed.), The Oxford handbook of philosophy of mathematics and logic (pp. 590–624). Oxford University Press.Google Scholar
  14. Feferman, S., & Strahm, T. (2000). The unfolding of non-finitist arithmetic. Annals of Pure and Applied Logic, 104(1–3), 75–96.Google Scholar
  15. Feferman, S., & Strahm, T. (2010). Unfolding finitist arithmetic. Review of Symbolic Logic, 3(4), 665–689.CrossRefGoogle Scholar
  16. Ferreira, F. (1990). Polynomial time computable arithmetic. In W. Sieg (Ed.), Logic and computation, proceedings of a workshop held at Carnegie Mellon University, 1987 vol. 106 of contemporary mathematics (pp. 137–156). Rhode Island: American Mathematical Society, Providence.Google Scholar
  17. Kahle, R. (2007). The Applicative Realm. Habilitation Thesis, Tübingen, 2007. Appeared in Textos de Mathemática 40, Departamento de Mathemática da Universidade de Coimbra, Portugal.Google Scholar
  18. Strahm, T. (2003). Theories with self-application and computational complexity. Information and Computation, 185, 263–297.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Institut für Informatik und angewandte MathematikUniversität BernBernSwitzerland

Personalised recommendations