# The Order of Reflection

## TL;DR

This paper characterizes the linear order of reflection patterns involving $oldsymbol{orall ext{-}}oldsymbol{ ext{and}}oldsymbol{ ext{-}}oldsymbol{ ext{exists}}$-reflection in descriptive set theory, establishing it as a well-ordered structure of length $oldsymbol{oldsymbol{ ext{omega}}^oldsymbol{ ext{omega}}}$.

## Contribution

It extends classical results by determining the order between all patterns of iterated $oldsymbol{ ext{Sigma}}^1_1$ and $oldsymbol{ ext{Pi}}^1_1$ reflection, revealing a complex prewellordering structure.

## Key findings

- The linear reflection order is a prewellordering of length ω^ω.
- Established relationships between linear and non-linear reflection patterns.
- Identified the structure of reflection patterns involving simultaneous $oldsymbol{ ext{Sigma}}^1_1$ and $oldsymbol{ ext{Pi}}^1_1$ reflection.

## Abstract

Extending Aanderaa's classical result that $\pi^1_1<\sigma^1_1$, we determine the order between any two patterns of iterated $\Sigma^1_1$- and $\Pi^1_1$-reflection. We show that this \emph{linear reflection order} is a prewellordering of length $\omega^\omega$. This requires considering the relationship between linear and some \emph{non-linear} reflection patterns, such as $\sigma^1_1\wedge\pi^1_1$, the pattern of simultaneous $\Sigma^1_1$- and $\Pi^1_1$-reflection.