Infinity

# Two Mathematicians Solved a Decades-Old Problem About Infinity with a Breakthrough Proof

Nothing will make you feel smaller or more mind-blown than the concept of infinity. Well, buckle up, because it only gets mind-blowier from there. Not only are there different kinds of infinity, they come in different sizes too. In 2016, two mathematicians blew the lids off their peers' heads by solving a decades-old problem about comparing infinities. Here is the 60-page proof summed up in just a few characters: p = t. Let us explain...

## Uncountable Apples and Oranges

Believe us when we say this article could be infinitely long, but ain't nobody got time for that. Let's start with infinity, a number that goes on and on and on forever. Because you could count 1, 2, 3 ... forever, there are infinite whole numbers. But, wait, you could do the same thing with just prime numbers. And even numbers too. Oh whoa. We just proved there are different kinds of infinity. Nice!

The numbers we were just talking about are called natural numbers, and they're just a tiny little branch on the overall number tree. If numbers are all one big umbrella, the category at the top that encompasses everything below is real numbers. A real number can be 3, or it can be √3, and it can even be 0.37846577246230456 — they're all real.

Comparing the infinite sets of natural numbers (whole numbers and zero, basically) is easy — there's a one-to-one relationship there. These are called countable sets because, uh, you can count them. Uncountable sets are what we're dealing with for real numbers. If you start at 1.0, is the next number in the infinite set 1.00001? Or is it 1.00000001? There are no spaces between numbers on the real number line, so they're uncountable, thus uncomparable to countable sets. Apples and oranges, people.

Every real number is essentially an infinity within itself, because you can have infinitely many decimal points. That being said, it's pretty clear to see that these overwhelmingly massive uncountable sets are larger than countable ones. This knowledge led mathematicians to wonder: if there are big and small infinite sets, can we have medium infinities too? Voilà! This question is the continuum hypothesis and it is literally one of the biggest (no pun intended) unsolved problems. In 1900, the German mathematician David Hilbert made a list of 23 of the most important problems in mathematics. He put the continuum hypothesis at the top.

Disproving the continuum hypothesis would mean that there are medium-sized infinities; proving it means there are only the bigs and the smalls. In 1940, mathematician Kurt Gödel showed that it couldn't be disproved within the usual axioms of mathematics, a.k.a. set theory. In the 1960s, mathematician Paul Cohen showed that the continuum hypothesis can't be proved by set theory. Ding ding ding! This won Cohen the Fields Medal, the highest honor in mathematics. And so, we inched just a tiny little bit closer to solving the continuum hypothesis.

## Eureka!

One particular infinity-related question has persisted since the 1940s, even after Gödel's and Cohen's work: The problem of p and t. Mathematicians believed that if we could crack this problem, we could once and for all solve that darn continuum hypothesis. And, phew, in 2016, the p and t problem was finally solved.

Enter our heroes: Maryanthe Malliaris, of the University of Chicago, and Saharon Shelah, of the Hebrew University of Jerusalem and Rutgers University. The two mathematicians published a proof to this problem in the Journal of the American Mathematical Society and were honored in July 2017 with one of the top prizes in the field of set theory. (Here's a much shorter summary of the proof by Cornell University's Justin Moore, by the way.) But what'd they solve?

The question at hand asks whether p (one variant of infinity) is equal to t (another variant of infinity). Both p and t quantify the minimum size of collections of subsets of the natural numbers in precise (and probably unique) ways. The details of p and t aren't important; just know this: Both sets are larger than the infinite set of natural numbers, and p is always less than or equal to t. If p is less than t, then p would be a medium infinity and the continuum hypothesis would be false. Pretty major.

In 2011, Malliaris and Shelah started working on a totally different problem. (It was about ordering problems based on complexity, building off Keisler's order, in case you were wondering.) In the process, they realized they were also, kind of, accidentally making headway with the p and t dilemma. So they went with it. The two published a 60-page paper that solved their initial problem and the famous p and t problem at the same time. By proving that p and t are equally complex, they concluded that p equals t.

They proved it by carving out their own lane between two branches of mathematics: set theory and model theory. Their work is already opening new frontiers of research in both fields. Why does that matter? The more we know about math, the more we can understand the mysterious ways of the world around us. Thanks, Malliaris and Shelah!

(Psst — There's an unsatisfying end to this breakthrough mathematics story, too: Their work didn't solve the continuum hypothesis like mathematicians thought it would. Oh well! However, experts are pretty sure there have to be medium-sized infinities. Infinity is so weird that even the weirdest theories could be true, probably, maybe. Why not?)

– Vi Hart

### Key Facts In This Video

1. The set of all real numbers includes every possible combination of digits extending infinitely among aleph null decimal places. 00:23

2. Here is Cantor's diagonal argument explained: 04:52

3. For every aleph number, there's infinite ordinal numbers. 08:52

Joanie Faletto  