END-EXTENSIONS OF MODELS OF WEAK ARITHMETIC FROM COMPLEXITY-THEORETIC CONTAINMENTS

We prove that if the linear-time and polynomial-time hierarchies coincide, then every model of Pi(1)(N) + inverted left perpendicular Omega(1) has a proper end-extension to a model of Pi(1)(N), and so Pi(1)(N) + inverted left perpendicular Omega(1) proves B Sigma(1). Under an even stronger complexity-theoretic assumption which nevertheless seems hard to disprove using present-day methods, Pi(1)(N) + inverted left perpendicularExp proves B Sigma(1). Both assumptions can be modified to versions which make it possible to replace Pi(1)(N) by I Delta(0) as the base theory. We also show that any proof that I Delta(0) + inverted left perpendicularExp does not prove a given finite fragment of B Sigma(1) has to be "nonrelativizing", in the sense that it will not work in the presence of an arbitrary oracle.