# How strong are single fixed points of normal functions?

## TL;DR

This paper explores the logical strength of fixed points of normal functions, establishing their equivalence to -induction, and extends previous work on derivatives of normal functions within reverse mathematics.

## Contribution

It demonstrates that the statement "every normal function has at least one fixed point" is equivalent to -induction, providing a new logical characterization in reverse mathematics.

## Key findings

- Fixed points of normal functions are equivalent to -induction.
- The paper extends previous results relating derivatives and -bar induction.
- Establishes a logical equivalence within the framework of reverse mathematics.

## Abstract

In a recent paper by M. Rathjen and the present author it has been shown that the statement ``every normal function has a derivative'' is equivalent to $\Pi^1_1$-bar induction. The equivalence was proved over $\mathbf{ACA_0}$, for a suitable representation of normal functions in terms of dilators. In the present paper we show that the statement ``every normal function has at least one fixed point'' is equivalent to $\Pi^1_1$-induction along the natural numbers.