For a theory T, we study relationships among I∆n+1(T), L∆n+1(T) and B * ∆ n+1 (T). These theories are obtained restricting the schemes of induction, minimization and (a version of) collection to ∆n+1(T) formulas. We obtain conditions on T (T is an extension of B * ∆ n+1 (T) or ∆ n+1 (T) is closed (in T) under bounded quantification) under which I∆ n+1 (T) and L∆ n+1 (T) are equivalent.These conditions depend on ThΠ n+2 (T), the Πn+2-consequences of T. The first condition is connected with descriptions of Th Π n+2 (T) as IΣ n plus a class of nondecreasing total Πn-functions, and the second one is related with the equivalence between ∆n+1(T)formulas and bounded formulas (of a language extending the language of Arithmetic). This last property is closely tied to a general version of a well known theorem of R. Parikh.Using what we call Π n -envelopes we give uniform descriptions of the previous classes of nondecreasing total Πn-functions. Πn-envelopes are a generalization of envelopes (see [10]) and are closely related to indicators (see [12]). Finally, we study the hierarchy of theories I∆n+1(IΣm), m ≥ n, and prove a hierarchy theorem.
Key words First order arithmetic, parameter free induction, exponentiation. MSC (2010) 03F30, 03H15We characterize the sets of all Π2 and all B(Σ1 ) (= Boolean combinations of Σ1 ) theorems of IΠ − 1 in terms of restricted exponentiation, and use these characterizations to prove that both sets are not deductively equivalent. We also discuss how these results generalize to n > 0. As an application, we prove that a conservation theorem of Beklemishev stating that IΠ − n + 1 is conservative over IΣ − n with respect to B(Σn + 1 ) sentences cannot be extended to Πn + 2 sentences.
In this paper we continue the study of the theories I 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 I n+1 (T) is n+2-axiomatizable. In particular, I n+1 (I n+1) gives an axiomatization of Th n 2 (I n+1) and is not finitely axiomatizable. This fact relates the fragment I n+1 (I n+1) + to induction rule for 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 n+2 and n+1 formulas (see [2]). In [7], this result is used to separate the fragments of Arithmetic introduced there: I n+1 (I n+1) and B * n+1 (I n+1). A basic result on n+1-induction rule is the following conservativeness theorem of C. Parsons (see [16] and 6.5): I n+1 is a n+2-conservative extension of
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 S i 2 and T 2 i and prove that they are ∀ i b conservative over their inference rule counterparts, and ∃∀ i b conservative over their parameter-free versions. A similar analysis of the i b -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.
We study the quantifier complexity and the relative strength of some fragments of arithmetic axiomatized by induction and minimization schemes for ∆n+1-formulas.Mathematics Subject Classification: 03F30, 03H15.
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