Archive for Mathematical Logic 43 (4):505-541 (2004)

Authors
Felicity Martin
University of Sydney
Alexis Fernandez
Florida International University
Abstract
For a theory T, we study relationships among IΔ n +1 (T), LΔ n+1 (T) and B * Δ n+1 (T). These theories are obtained restricting the schemes of induction, minimization and (a version of) collection to Δ n+1 (T) formulas. We obtain conditions on T (T is an extension of B * Δ n+1 (T) or Δ n+1 (T) is closed (in T) under bounded quantification) under which IΔ n+1 (T) and LΔ n+1 (T) are equivalent. These conditions depend on Th Πn +2 (T), the Π n+2 –consequences of T. The first condition is connected with descriptions of Th Πn +2 (T) as IΣ n plus a class of nondecreasing total Π n –functions, and the second one is related with the equivalence between Δ 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 Π n –functions. Π n –envelopes are a generalization of envelopes (see [10]) and are closely related to indicators (see [12]). Finally, we study the hierarchy of theories IΔ n+1 (IΣ m ), m≥n, and prove a hierarchy theorem
Keywords Induction  Collection  formulas
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Reprint years 2004
DOI 10.1007/s00153-003-0197-8
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References found in this work BETA

On the Induction Schema for Decidable Predicates.Lev D. Beklemishev - 2003 - Journal of Symbolic Logic 68 (1):17-34.
On Σ1‐Definable Functions Provably Total in I ∏ 1−.Teresa Bigorajska - 1995 - Mathematical Logic Quarterly 41 (1):135-137.

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