**April 20**

Andrés Cordón Franco,
Universidad de Sevilla

**Induction and collection up to definable elements: calibrating the strength of parameter-free $\Delta_n$-minimization.**

In this talk we shall deal with fragments of first-order Peano Arithmetic obtained by restricting the conclusion of the induction or the collection axiom to elements in a prescribed subclass $D$ of the universe. Fix $n>0$. The schemes of $\Sigma_n$-induction up to $\Sigma_m$-definable elements and the schemes of $\Sigma_n$-collection up to $\Sigma_m$-definable elements form two families of subtheories of $I\Sigma_n$ and $B\Sigma_n$, respectively, obtained in this way.

The properties of $\Sigma_n$-induction up to $\Sigma_m$-definable elements for $n\geq m$ are reasonably well understood and interesting applications of these fragments are known. However, an analysis of the case $n<m$ was pending. In the first part of this talk, we address this problem and show that it is related to the following general question: 'Under which conditions on a model $M$ can we prove that every non-empty $\Sigma_m$-definable subset of $M$ contains some $\Sigma_m$-definable element?'

In the second part of the talk, we show that, for each $n\geq 1$, the scheme of $\Sigma_n$-collection up to $\Sigma_n$-definable elements provides us with an axiomatization of the $\Sigma_{n+1}$-consequences of $B\Sigma_n$. As an application, we obtain that $B\Sigma_n$ is $\Sigma_{n+1}$-conservative over parameter-free $\Delta_n$-minimization (plus $I\Sigma_{n-1}$), thus partially answering a question of R. Kaye.

This is joint work with F.Félix Lara-Martín (University of Seville).