## Journal of Symbolic Logic

### Finite Injury and $\sum_1$-Induction

Michael Mytilinaios

#### 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 $\sigma_1$-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.

#### Article information

Source
J. Symbolic Logic, Volume 54, Issue 1 (1989), 38-49.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183742849

Mathematical Reviews number (MathSciNet)
MR987320

Zentralblatt MATH identifier
0671.03029

JSTOR
Mytilinaios, Michael. Finite Injury and $\sum_1$-Induction. J. Symbolic Logic 54 (1989), no. 1, 38--49. https://projecteuclid.org/euclid.jsl/1183742849