Two infinite quantities and their surprising relationship

TL;DR

This paper discusses the surprising equivalence of two infinite quantities, r and t, in set theory, highlighting recent breakthroughs by Malliaris and Shelah that resolve longstanding questions about infinite set sizes.

Contribution

The paper explains the recent proof that two well-studied infinite quantities, r and t, are equal, advancing understanding in set theory and infinite set classification.

Findings

r and t are proven to be equal.

The work connects model theory with set theory.

It highlights recent breakthroughs in infinite set classification.

Abstract

As early as the 17th century, Galileo Galilei wondered how to compare the sizes of infinite sets. Fast forward almost four hundred years, and in the summer of 2017, at the 6th European Set Theory Conference, a young model theorist, Maryanthe Malliaris, and the well-known polymath, Saharon Shelah, received the Hausdorff Medal for the most influential work in set theory published in the last five years. Malliaris and Shelah made significant breakthroughs both regarding a model theoretic classification problem (that is, sorting certain objects into types), and proved that two well-studied infinite quantities, $\mathfrak{p}$ and $\mathfrak{t}$, are in fact the same. This latter result is the focus of our expository paper.