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 $\sigma_1$-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 $\Sigma_1$-induction.