Local induction and provably total computable functions

Let I¦− 2 denote the fragment of Peano Arithmetic obtained by restricting the induction scheme to parameter free ¦2 formulas. Answering a question of R. Kaye, L. Beklemishev showed that the provably total computable functions of I¦− 2 are, precisely, the primitive recursive ones. In this work we give a new proof of this fact through an analysis of certain local variants of induction principles closely related to I¦− 2 . In this way, we obtain a more direct answer to Kaye’s question, avoiding the metamathematical machinery (reflection principles, provability logic,...) needed for Beklemishev’s original proof. Our methods are model–theoretic and allow for a general study of I¦− n+1 for all n ¸ 0. In particular, we derive a new conservation result for these theories, namely that I¦− n+1 is ¦n+2–conservative over I§n for each n ¸ 1.Ministerio de Ciencia e Innovación MTM2008–06435Ministerio de Ciencia e Innovación MTM2011–2684

Local Induction and Provably Total Computable Functions: A Case Study

Let IΠ−2 denote the fragment of Peano Arithmetic obtained by restricting the induction scheme to parameter free Π2 formulas. Answering a question of R. Kaye, L. Beklemishev showed that the provably total computable functions (p.t.c.f.) of IΠ−2 are, precisely, the primitive recursive ones. In this work we give a new proof of this fact through an analysis of the p.t.c.f. of certain local versions of induction principles closely related to IΠ−2 . This analysis is essentially based on the equivalence between local induction rules and restricted forms of iteration. In this way, we obtain a more direct answer to Kaye’s question, avoiding the metamathematical machinery (reflection principles, provability logic,...) needed for Beklemishev’s original proof.Ministerio de Ciencia e Innovación MTM2008–0643

Existentially Closed Models in the Framework of Arithmetic

We prove that the standard cut is definable in each existentially closed model of IΔ0 + exp by a (parameter free) П1–formula. This definition is optimal with respect to quantifier complexity and allows us to improve some previously known results on existentially closed models of fragments of arithmetic.Ministerio de Educación y Ciencia MTM2011–2684

On Rules and Parameter Free Systems in Bounded Arithmetic

We present model–theoretic techniques to obtain conservation results for first order bounded arithmetic theories, based on a hierarchical version of the well known notion of an existentially closed model.Ministerio de Educación y Ciencia MTM2005-0865

Existentially Closed Models and Conservation Results in Bounded Arithmetic

We develop model-theoretic techniques to obtain conservation results for first order Bounded Arithmetic theories, based on a hierarchical version of the well-known notion of an existentially closed model. We focus on the classical Buss' theories Si2 and Ti2 and prove that they are ∀Σbi conservative over their inference rule counterparts, and ∃∀Σbi conservative over their parameter-free versions. A similar analysis of the Σbi-replacement scheme is also developed. The proof method is essentially the same for all the schemes we deal with and shows that these conservation results between schemes and inference rules do not depend on the specific combinatorial or arithmetical content of those schemes. We show that similar conservation results can be derived, in a very general setting, for every scheme enjoying some syntactical (or logical) properties common to both the induction and replacement schemes. Hence, previous conservation results for induction and replacement can be also obtained as corollaries of these more general results.Ministerio de Educación y Ciencia MTM2005-08658Junta de Andalucía TIC-13

Fragments of Arithmetic and true sentences

By a theorem of R. Kaye, J. Paris and C. Dimitracopoulos, the class of the ¦n+1–sentences true in the standard model is the only (up to deductive equivalence) consistent ¦n+1–theory which extends the scheme of induction for parameter free ¦n+1–formulas. Motivated by this result, we present a systematic study of extensions of bounded quantifier complexity of fragments of first–order Peano Arithmetic. Here, we improve that result and show that this property describes a general phenomenon valid for parameter free schemes. As a consequence, we obtain results on the quantifier complexity, (non)finite axiomatizability and relative strength of schemes for ¢n+1–formulas.Junta de Andalucía TIC-13

On the quantifier complexity of Δ n+1 (T)– induction

In this paper we continue the study of the theories IΔ n+1 (T), initiated in [7]. We focus on the quantifier complexity of these fragments and theirs (non)finite axiomatization. A characterization is obtained for the class of theories such that IΔ n+1 (T) is Π n+2 –axiomatizable. In particular, IΔ n+1 (IΔ n+1 ) gives an axiomatization of Th Π n+2 (IΔ n+1 ) and is not finitely axiomatizable. This fact relates the fragment IΔ n+1 (IΔ n+1 ) to induction rule for Δ n+1 –formulas. Our arguments, involving a construction due to R. Kaye (see [9]), provide proofs of Parsons’ conservativeness theorem (see [16]) and (a weak version) of a result of L.D. Beklemishev on unnested applications of induction rules for Π n+2 and Δ n+1 formulas (see [2]).Ministerio de Educación y Cultura DGES PB96-134

Envelopes, indicators and conservativeness

A well known theorem proved (independently) by J. Paris and H. Friedman states that BΣn +1 (the fragment of Arithmetic given by the collection scheme restricted to Σn +1‐formulas) is a Πn +2‐conservative extension of IΣn (the fragment given by the induction scheme restricted to Σn ‐formulas). In this paper, as a continuation of our previous work on collection schemes for Δn +1(T )‐formulas (see [4]), we study a general version of this theorem and characterize theories T such that T + BΣn +1 is a Πn +2‐conservative extension of T . We prove that this conservativeness property is equivalent to a model‐theoretic property relating Πn ‐envelopes and Πn ‐indicators for T . The analysis of Σn +1‐collection we develop here is also applied to Σn +1‐induction using Parsons' conservativeness theorem instead of Friedman‐Paris' theorem. As a corollary, our work provides new model‐theoretic proofs of two theorems of R. Kaye, J. Paris and C. Dimitracopoulos (see [8]): BΣn +1 and IΣn +1 are Σn +3‐conservative extensions of their parameter free versions, BΣ–n +1 and IΣ–n +1.Junta de Andalucía TIC-13

Provably Total Primitive Recursive Functions: Theories with Induction

A natural example of a function algebra is R (T), the class of provably total computable functions (p.t.c.f.) of a theory T in the language of first order Arithmetic. In this paper a simple characterization of that kind of function algebras is obtained. This provides a useful tool for studying the class of primitive recursive functions in R (T). We prove that this is the class of p.t.c.f. of the theory axiomatized by the induction scheme restricted to (parameter free) Δ1(T)–formulas (i.e. Σ1–formulas which are equivalent in T to Π1–formulas). Moreover, if T is a sound theory and proves that exponentiation is a total function, we characterize the class of primitive recursive functions in R (T) as a function algebra described in terms of bounded recursion (and composition). Extensions of this result are related to open problems on complexity classes. We also discuss an application to the problem on the equivalence between (parameter free) Σ1–collection and (uniform) Δ1–induction schemes in Arithmetic. The proofs lean upon axiomatization and conservativeness properties of the scheme of Δ1(T)–induction and its parameter free version