| At the turn of the 20th century, mathematics seemed to be on a perfect path. After reckoning for decades with imprecise definitions and disturbing paradoxes (in a period now known as the foundational crisis of mathematics), mathematicians had finally started to establish a sturdy foundation for the field. Their goal was to develop a minimal list of basic truths, or axioms, upon which all of mathematics could rest. If they succeeded, it would be possible to derive even the most esoteric results from these simple, natural assumptions. First, mathematicians could use the axioms to prove straightforward statements; then they could use those statements to prove even more complicated theorems; and so on. Future mathematicians could continue to add to this edifice. With enough time and creativity, no truth would be out of reach.
Or so everyone thought.
It turned out that the dream of a "complete" mathematics, championed most prominently by the mathematician David Hilbert, represented an optimistic, Enlightenment-era view of the matter. In 1931, Kurt Gödel crushed Hilbert's dream with proofs of his two "incompleteness theorems." The first showed that, for any sufficiently powerful set of axioms (including those that underlie modern mathematics), there are some mathematical truths that cannot be proved true or false. They're provably unprovable. Gödel's second proof built on this idea, demonstrating that such a set of axioms can never be used to prove its own consistency. The axioms might lead to contradictory statements, and mathematicians would be none the wiser if they were limited to using just those axioms.
You might expect such results to stifle progress in a discipline so committed to the pursuit of absolute truth. But most mathematicians are still able to prove the statements they want to prove. Meanwhile, Gödel's incompleteness theorems opened a whole new realm of inquiry within math. Mathematicians could now focus not just on finding out what was true, but on finding out what was knowable. Grappling with this idea remains a central practice within mathematics today.
What's New and Noteworthy Thanks to Gödel's work, mathematicians know that if they start with the usual axioms that underlie modern mathematics, there will be statements they can't prove true or false. To prove such statements, they need to introduce new axioms to their foundation. These new axioms often deal with a counterintuitive and fascinating mathematical concept: infinity.
Infinity comes in many different sizes. The set of natural numbers (0, 1, 2, 3, 4 …) is equal in size to the set of all even numbers, and to the set of all fractions. But the set of all real numbers (all the numbers on the number line) is larger than these sets, even though they're all infinite. Mathematicians have come up with axioms to define far larger, and more exotic, types of infinity. Recently, Quanta covered work in which three mathematicians claimed to have discovered two new kinds of infinity that don't behave as mathematicians expect. The finding might mean that the mathematical universe is far more chaotic than anticipated.
In general, the best way to define new axioms — that is, the best way to inch toward a more complete (though never entirely complete) understanding of the world of mathematics — is hotly debated.
A few years after Gödel published his incompleteness theorems, Alan Turing and others built on his ideas, showing that there are mathematical statements that are "undecidable." They can't be solved by any computer algorithm.
Just as Hilbert's dream wasn't safe from the notion of incompleteness, it wasn't safe from the notion of undecidability, either. In 1970, mathematicians proved that another major pillar of Hilbert's program — his "10th problem," one of a list of 23 questions he'd proposed to help reinforce mathematics — was undecidable. The problem had challenged mathematicians to find an algorithm that could determine whether any "Diophantine" equation (a class of simple polynomials, such as y = x2 – 1) has whole-number solutions. This year I reported on work that significantly extended the scope of this result — a demonstration that yet another part of the mathematical world is unknowable. My favorite (potentially) unanswerable question from Quanta's coverage: Can a set of shapes be duplicated to cover all of space? The hunt for such arrangements, called "tilings," is a major topic in mathematics. And it turns out that some tiling problems are undecidable: Given some sets of shapes, it's not possible to figure out whether they'll tile a space. But what about when you consider just one shape? Could a statement about a single tile be undecidable? In 2022, Rachel Greenfeld and Terence Tao shocked the tiling part of the math world when they found a single shape that can tile all of space in a non-repeating, or aperiodic, pattern. But just because an aperiodic tile exists doesn't mean that an undecidable one does. The two mathematicians hope that the techniques they developed might help them eventually find a single undecidable tile, too. Far from the death knell it may have initially seemed, incompleteness thus has become a boon to mathematics. The only thing more intriguing than pure, incontrovertible truth, it turns out, is the knowledge that such truth is often unattainable. Future mathematicians won't just have the job of perpetually adding layers to the edifice of knowledge, as Hilbert envisioned. Their task will be much more difficult: to decide for us what things can and cannot be decided. | 
