Existentially Closed Models and Conservation Results in Bounded Arithmetic

Abstract

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.

Induction rules in bounded arithmetic

“…We start with the easier, and already understood, case ofΣ b i rules. The conservation result forΣ b i -(P )IND R below, which also implies a conservation result forΣ b i -(P )IND − , was proved by Cordón-Franco, Fernández-Margarit, and Lara-Martin [18] by model-theoretic means. It generalizes the special case for T ⊆ ∀Σ b i shown proof-theoretically by Bloch [6]; an analogous result for IE − n was shown earlier by Kaye [28].…”

“…A crucial property is that induction rules are equivalent to their parameter-free versions. The case of Σb i was already proved in [18], but we include it for completeness anyway.…”

“…We tried our best to conduct an in-depth examination of parameter-free and inference-rule versions of the IND and PIND schemes, that also applies, by the results of Section 4, to their common variants like LIND and minimization schemes. However, we left out other schemes of interest in bounded arithmetic: in particular, the choice (aka replacement or bounded collection) scheme BB (which was studied in [18]), and analogues of LIND with induction up to bounds given by more general classes of terms (including LLIND, etc.). Related to BB , we might be interested in variants of (P )IND and other schemes for the non-strict Σ b i and Π b i formula classes: it is well known that with parameters, the strict and non-strict (P )IND schemes are equivalent-both define the familiar theories S i 2 and T i 2 .…”

“…In contrast to all these results, much less is known about parameter-free induction axioms and induction rules in the context of bounded arithmetic: the early work of Kaye [28] introduced the parameter-free subtheories IE − i of I∆ 0 , while the only investigation of parameterfree Buss's theories was done by Bloch [6], who studied proof-theoretically Σ b i parameter-free induction rules 1 in a sequent formalism, and Cordón-Franco, Fernandéz-Margarit, and Lara-Martín [18], whose main results concern conservativity of the theories S i 2 and T i 2 over the parameter-free and induction-rule versions ofΣ b i -PIND andΣ b i -IND, and conservativity of BBΣ b i over its rule version. They rely on model-theoretic methods exploiting variants of existentially closed models.…”

“…The most substantial technical part of the paper comes in Section 5, which is devoted to conservation results. We recall the conservation of T i 2 and S i 2 over Σb i -(P )IND R (Theorem 5.1) from [6,18], and we set out to prove an analogous conservation result over Πb i -(P )IND R (Theorem 5.9). A key part of the proof is a new witnessing theorem for ∀∃∀ Σb i−1 consequences (and ∀∃∀ Σb i consequences) of T i 2 and S i 2 , which may be of independent interest (Theorem 5.4 and Proposition 5.5).…”