Finite injury and <i>Σ</i>₁-induction

Abstract

Abstract Working in the language of first-order arithmetic we consider models of the base theory P − . Suppose M is a model of P − and let M satisfy induction for Σ 1 -formulas. First it is shown that the Friedberg-Muchnik finite injury argument can be performed inside M , and then, using a blocking method for the requirements, we prove that the Sacks splitting construction can be done in M . So, the “amount” of induction needed to perform the known finite injury priority arguments is Σ 1 -induction.