Abstract Let T be a second-order arithmetical theory, Λ a well-order, λ < Λ and X ⊆ ℕ. We use $[\lambda |X]_T^{\rm{\Lambda }}\varphi$ as a formalization of “φ is provable from T and an oracle for the set X, using ω-rules of nesting depth at most λ”. For a set of formulas Γ, define predicative oracle reflection for T over Γ (Pred–O–RFNΓ(T)) to be the schema that asserts that, if X ⊆ ℕ, Λ is a well-order and φ ∈ Γ, then $$\forall \,\lambda < {\rm{\Lambda }}\,([\lambda |X]_T^{\rm{\Lambda }}\varphi \to \varphi ).$$ In particular, define predicative oracle consistency (Pred–O–Cons(T)) as Pred–O–RFN{0=1}(T). Our main result is as follows. Let ATR0 be the second-order theory of Arithmetical Transfinite Recursion, ${\rm{RCA}}_0^{\rm{*}}$ be Weakened Recursive Comprehension and ACA be Arithmetical Comprehension with Full Induction. Then, $${\rm{ATR}}_0 \equiv {\rm{RCA}}_0^{\rm{*}} + {\rm{Pred - O - Cons\ }}\left( {{\rm{RCA}}_0^{\rm{*}} } \right) \equiv {\rm{RCA}}_0^{\rm{*}} + \,{\rm{Pred - O - Cons\ }}\left( {{\rm{RCA}}_0^{\rm{*}} } \right) \equiv {\rm{RCA}}_0^{\rm{*}} + \,{\rm{Pred - O - RFN}}\,_{{\bf{\Pi }}_2^1 } \left( {{\rm{ACA}}} \right).$$ We may even replace ${\rm{RCA}}_0^{\rm{*}}$ by the weaker ECA0, the second-order analogue of Elementary Arithmetic. Thus we characterize ATR0, a theory often considered to embody Predicative Reductionism, in terms of strong reflection and consistency principles.

Abstract We prove that the standard cut is definable in each existentially closed model ofIΔ0+ 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.

Abstract LetTbe a recursively enumerable theory extending Elementary Arithmetic EA. L. D. Beklemishev proved that the Σ2local reflection principle forT, (T), is conservative over the Σ1local reflection principle, (T), with respect to boolean combinations of Σ1-sentences; and asked whether this result is best possible. In this work we answer Beklemishev's question by showing that Π2-sentences are not conserved forT= EA + “f is total,” wherefis any nondecreasing computable function with elementary graph. We also discuss how this result generalizes ton> 0 and obtain as an application that forn> 0, is conservative overIΣnwith respect to Πn+2-sentences.

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.

Abstract LetTbe a recursive theory in the language of first order Arithmetic. We prove that ifTextends: (a) the scheme of parameter free Δ1-minimization (plusexp). or (b) the scheme of parameter free Π1-induction, then there are no Σ1-maximal models with respect toT. As a consequence, we obtain a new proof of an unpublished theorem of Jeff Paris stating that Σ1-maximal models with respect toIΔ0+expdo not satisfy the scheme of Σ1-collectionBΣ1.