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

Abstract

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

Fragments of Arithmetic and true sentences

“…To this end, we first establish a result relating the syntactical complexity of a theory with the rate of growth of the recursive functions which are provably total in that theory. Proposition 4.8 below extends Claim 2.5.1 in [8] and generalizes theorems of R. Parikh and Bigorajska [5] on recursive functions provably total in I∆ 0 and IΠ − 1 , respectively. Proposition 4.8 Let T be a theory and ϕ(x, y) ∈ Σ 1 such that T + BΣ 1 ∀x∃y ϕ(x, y).…”

“…Moreover, in [8] for each n ≥ 1 it is presented a Π n -formula, y = K z n (x), which expresses the iteration of the function y = K n (x) and it is established that (see Section 3 in [8] for details):…”

“…In [8] it is also proved that IΣ − n+1 ∀x∃y (y = K x+1 n (x + 2)). Hence, an argument similar to that of part 1. of Theorem 4.9 shows that there is no class Γ of Σ n+2 -sentences such that BΣ n+1 + Γ is a consistent extension of IΣ − n+1 .…”

A note on parameter free Π1-induction and restricted exponentiation

“…Secondly, it follows from Theorem 4.7 and Corollary 3.6 that Th Π2 (IΠ − 1 ) proves ∀x ∃!y (y = g D (x)) but does not prove ∃u ∀x > u ∃y ≤ t(x) (y = g D (x)) for any term t(x). Thus, we have: [3], where the authors studied the quantifier complexity of the induction schema for the class of Δ n +1 formulas in an arithmetic theory T , IΔ n +1 (T ). Whereas for n > 0 this scheme is not Σ n +2 axiomatizable for any theory T , for n = 0 the authors gave a number of examples for which IΔ 1 (T ) is Σ 2 axiomatizable (e.g., T = IΔ 0 , or BΣ 1 ).…”

“…Question 4.10 (Problem 7.2 of [3]) Suppose IΔ 0 ⊆ T ⊆ IΣ 1 and T is closed under Σ 1 collection rule. Are the following conditions equivalent?…”