Can something finite exist in infinity

Math sensation: from infinity to infinity

What are the characteristics of a complex theory? This is not an easy question and is still a topical research topic. Keisler describes the complexity of a theory by the range of what can happen in the theory - theories in which more can happen are more complex than those in which hardly anything happens.

A little over a decade after Keisler introduced his order, Shelah published a book in which he showed that there is a demarcation between the complexity classes - like a dividing line that separates more complex theories from simpler ones. In the next 30 years, however, there was little progress in this area.

Comeback of the Keisler order

But in 2009 the Keisler order experienced its comeback: Malliaris studied Keisler's work during her doctoral thesis and her subsequent publications. In 2011 she and Shelah started working together in this area. For example, you wanted to understand what transforms a simple theory into a maximally complex theory: Which properties make theories complex?

You already knew then that ordered mathematical theories were extremely complex. So if theories contain something like a greater or less than sign, they are maximally complex. Malliaris and Shelah wanted to find out whether a weakened kind of order also leads to a theory becoming maximally complex.

Infinite levels of complexity

In the course of their research, Malliaris and Shelah discovered an unexpected connection: If both order and its weakened variant are maximally complex, that would also mean that p and t are the same. They showed that two different areas of mathematics are much more closely related than previously assumed.

In 2016, Malliaris and Shelah published a 60-page paper that solved both problems: They proved that both order properties are equally complex, and that p and t to match. In another work they also showed that Keisler's order not only has two, but - as Keisler already suspected - an infinite number of levels of complexity. "Somehow one thing suddenly led to another," says Malliaris. "We managed to solve a lot of different problems."

Last July Malliaris and Shelah received the Hausdorff Medal, one of the most important prizes in the field of set theory. This honor reflects their unexpected and powerful evidence. Most mathematicians expected that p would be smaller than t and their relationship to one another would therefore not be provable. But Malliaris and Shelah proved: Both infinities are the same size. However, their work also showed that this question has much more depth than mathematicians suspected.

Building bridges between the research areas

“Almost everyone assumed that there was evidence of the equality of p and t Surprisingly, it would be based on a short and clever argument rather than a complicated method, ”notes Justin Moore, a mathematician at Cornell University.

Instead, Malliaris and Shelah proved that p and t are equal by showing a previously unknown connection between model theory and set theory. This should open up exciting new opportunities in both areas. The work also eliminates a question mathematicians hoped would help solve the continuum hypothesis. As before, almost all experts assume that the unprovable conjecture is wrong: It would be very unusual if there were only the two sizes of infinity that have been found so far.