Plus ultra

TL;DR

This paper introduces the class of tame ultraimaginaries in simple theories, explores their properties, and establishes results on their elimination and canonical base behavior.

Contribution

It defines tame ultraimaginaries and proves basic simplicity-theoretic properties, including feeble elimination and uniform canonical base results.

Findings

Feeble elimination of supersimple ultraimaginaries up to certain ranks

Introduction of tame ultraimaginaries with well-behaved properties

Uniform versions of the weak canonical base property

Abstract

We define a reasonably well-behaved class of ultraimaginaries, i.e.\ classes modulo invariant equivalence relations, called {\em tame}, and establish some basic simplicity-theoretic facts. We also show feeble elimination of supersimple ultraimaginaries: If $e$ is an ultraimaginary definable over a tuple $a$ with $SU(a)<\omega^{\alpha+1}$, then $e$ is eliminable up to rank $<\omega^\alpha$. Finally, we prove some uniform versions of the weak canonical base property.