We investigate the theory IΔ 0 + Ω 1 and strengthen [Bu86. Theorem 8.6] to the following: if NP ≠ co-NP. then Σ-completeness for witness comparison formulas is not provable in bounded arithmetic. i.e. $I\delta_0 + \Omega_1 + \nvdash \forall b \forall c (\exists a(\operatorname{Prf}(a.c) \wedge \forall = \leq a \neg \operatorname{Prf} (z.b))\\ \rightarrow \operatorname{Prov} (\ulcorner \exists a(\operatorname{Prf}(a. \bar{c}) \wedge \forall z \leq a \neg \operatorname{Prf}(z.\bar{b})) \urcorner)).$ Next we study a "small reflection principle" in bounded arithmetic. We prove that for (...) all sentences φ $I\Delta_0 + \Omega_1 \vdash \forall x \operatorname{Prov}(\ulcorner \forall y \leq \bar{x} (\operatorname{Prf} (y. \overline{\ulcorner \varphi \urcorner}) \rightarrow \varphi)\urcorner).$ The proof hinges on the use of definable cuts and partial satisfaction predicates akin to those introduced by Pudlak in [Pu86]. Finally, we give some applications of the small reflection principle, showing that the principle can sometimes be invoked in order to circumvent the use of provable Σ-completeness for witness comparison formulas. (shrink)

V i 2 ⊢A iff for some term t :S i 2 ⊢ “2 i exists→ A”, a bounded first-order formula, i ≥1. V i 2 is not Π b 1 -conservative over S i 2 . Any model of V 2 not satisfying Exp satisfies the collection scheme BΣ 0 1 . V 1 3 is not Π b 1 -conservative over S 2.

We define and study a new restricted consistency notion RCon ∗ for bounded arithmetic theories T 2 j . It is the strongest ∀ Π 1 b -statement over S 2 1 provable in T 2 j , similar to Con in Krajíček and Pudlák, 29) or RCon in Krajı́ček and Takeuti 107). The advantage of our notion over the others is that RCon ∗ can directly be used to construct models of T 2 j . We apply this by (...) proving preservation theorems for theories of bounded arithmetic of the following well-known kind: The ∀ Π 1 b -separation of bounded arithmetic theories S 2 i from T 2 j is equivalent to the existence of a model of S 2 i which does not have a Δ 0 b -elementary extension to a model of T 2 j . More specific, let M ⊨ Ω 1 nst denote that there is a nonstandard element c in M such that the function n ↦ 2 log c is total in M . Let BLΣ 1 b be the bounded collection schema for Σ 1 b -formulas. We obtain the following preservation results: the ∀ Π 1 b -separation of S 2 i from T 2 j is equivalent to the existence of 1. a model of S 2 i +Ω 1 nst which is 1 b -closed w.r.t. T 2 j , 2. a countable model of S 2 i + BLΣ 1 b without weak end extensions to models of T 2 j. (shrink)