Semi-honest subrecursive degrees and the collection rule in arithmetic
Tóm tắt
By a result of L.D. Beklemishev, the hierarchy of nested applications of the
$$\Sigma _1$$
-collection rule over any
$$\Pi _2$$
-axiomatizable base theory extending Elementary Arithmetic collapses to its first level. We prove that this result cannot in general be extended to base theories of arbitrary quantifier complexity. In fact, given any recursively enumerable set of true
$$\Pi _2$$
-sentences, S, we construct a sound
$$(\Sigma _2 \! \vee \! \Pi _2)$$
-axiomatized theory T extending S such that the hierarchy of nested applications of the
$$\Sigma _1$$
-collection rule over T is proper. Our construction uses some results on subrecursive degree theory obtained by L. Kristiansen.
Tài liệu tham khảo
Adamowicz, Z., Bigorajska, T.: Existentially closed structures and Gödel’s second incompleteness theorem. J. Symb. Logic 66, 349–356 (2001)
Avigad, J.: Saturated models of universal theories. Ann. Pure Appl. Logic 118, 219–234 (2002)
Basu, S.: On the structure of subrecursive degrees. J. Comput. Syst. Sci. 4, 452–464 (1970)
Beklemishev, L.D.: Induction rules, reflection principles, and provably recursive functions. Ann. Pure Appl. Logic 85, 193–242 (1997)
Beklemishev, L.D.: A proof-theoretic analysis of collection. Arch. Math. Logic 37, 275–296 (1998)
Beklemishev, L.D.: On the Induction Schema for Decidable Predicates. J. Symb. Logic 68, 17–34 (2003)
Beklemishev, L.D.: Provability algebras and proof-theoretic ordinals. I. Ann. Pure Appl. Logic 128, 103–123 (2004)
Beklemishev, L.D.: Reflection principles and provability algebras in formal arithmetic. Russ. Math. Surv. 60, 197–268 (2005)
Cai, M.: Degrees of relative provability. Notre Dame J. Formal Logic 53, 479–489 (2012)
Cordón-Franco, A., Fernández-Margarit, A., Lara-Martín, F.F.: Provably Total Primitive Recursive Functions: Theories With Induction. In: Marcinkowski, J., Tarlecki, A. (eds.) Computer Science Logic. CSL 2004. Lecture Notes in Computer Science, vol. 3210, Springer, Berlin, Heidelberg (2004)
Cordón-Franco, A., Fernández-Margarit, A., Lara-Martín, F.F.: Fragments of Arithmetic and true sentences. Math. Log. Quart. 51, 313–328 (2005)
Cordón-Franco, A., Fernández-Margarit, A., Lara-Martín, F.F.: Envelopes, indicators and conservativeness. Math. Log. Quart. 52, 51–70 (2006)
Cordón-Franco, A., Fernández-Margarit, A., Lara-Martín, F.F.: On axiom schemes for \(T\)-provably \(\Delta _1\) formulas. Arch. Math. Logic 53, 327–349 (2014)
Hájek, P., Pudlák, P.: Metamathematics of First-Order Arithmetic. Springer, Berlin, Heidelberg (1993)
Kreisel, G., Lévy, A.: Reflection Principles and Their Use for Establishing the Complexity of Axiomatic Systems. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 14, 97–142 (1968)
Kristiansen, L.: Papers on subrecursion theory, Dr Scient Thesis, ISSN 0806-3036, ISBN 82-7368-130-0, Research report 217, Department of Informatics, University of Oslo (1996)
Kristiansen, L.: A jump operator on honest subrecursive degrees. Arch. Math. Logic 37, 105–125 (1998)
Kristiansen, L.: Subrecursive degrees and fragments of Peano Arithmetic. Arch. Math. Logic 40, 365–397 (2001)
Kristiansen, L.: Degrees of total algorithms versus degrees of honest functions. In: Cooper, S.B., Dawar, A., Löwe, B. (eds.) How the World Computes. CiE 2012. Lecture Notes in Computer Science, vol. 7318, Springer, Berlin, Heidelberg (2012)
Kristiansen, L., Schlage-Puchta, J.C., Weiermann, A.: Streamlined subrecursive degree theory. Ann. Pure Appl. Logic 163, 698–716 (2012)
Machtey, M.: Augmented loop languages and classes of computable functions. J. Comput. Syst. Sci. 6, 603–624 (1972)
Machtey, M.: The honest subrecursive classes are a lattice. Inf. Control 24, 247–263 (1974)
Machtey, M.: On the density of honest subrecursive classes. J. Comput. Syst. Sci. 10, 183–199 (1975)
Meyer, A.R., Ritchie, D.M.: A classification of the recursive functions. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik 18, 71–82 (1972)
Odifreddi, P.: Classical Recursion Theory. North-Holland (1989)
Rose, H.E.: Subrecursion. Functions and Hierarchies. Clarendon Press, Oxford (1984)