Preservation theorems and restricted consistency statements in bounded arithmetic

Abstract

We define and study a new restricted consistency notion RCon ∗(T 2 j) 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( G i ) in Krajíček and Pudlák, (Z. Math. Logik Grundl. Math. 36 (1990) 29) or RCon( T i 1) in Krajı́ček and Takeuti (Ann. Math. Artificial Intelligence 6 (1992) 107). The advantage of our notion over the others is that RCon ∗(T 2 j) 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 (1⩽ i⩽ 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(n) 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 (1⩽ i⩽ 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 .