Joel David Hamkins is a mathematician and philosopher specializing in set theory, the foundations of mathematics and the nature of infinity. He has the highest score on MathOverflow, the question-and-answer platform for research mathematicians, with more than 246,000 points, and maintains the blog infinitelymore.xyz.
In his conversation with Lex Fridman, he recounts the history of infinity from Aristotle and Galileo to Cantor, Gödel and Cohen, and presents some of the deepest ideas in modern mathematics.
Cantor's central idea, that some infinities are larger than others, shook mathematics in the late 19th century. As Hamkins explains, it created a theological crisis—since infinity was associated with God—a mathematical civil war with Kronecker, and a psychological breakdown for Cantor himself, who spent the final years of his life in sanatoriums.
Galileo had already observed the paradox that square numbers can be matched one-to-one with all numbers, despite the gaps between them, and had remained puzzled.
The modern approach rests on the Cantor–Hume principle: two collections have the same size if and only if there is a one-to-one correspondence between them. This means that line segments of different lengths have the same number of points, and that two circles, however different they may be, have exactly the same points. This conflicts with Euclid's principle that the whole is always greater than the part, and this tension was fully resolved only with Cantor.
Hamkins uses the example of Hilbert's hotel to explain countable infinity. A hotel with infinitely many rooms can be full and still accommodate new guests by moving the existing guests one room along. Even more impressively, it can accommodate Hilbert's bus with infinitely many seats, or even Hilbert's train with infinitely many carriages, each with infinitely many seats—all without making the infinity any larger. This technique shows that the union of countably many countable sets remains countable.
The rational numbers, although densely ordered—between any two rational numbers there is another rational number—are also only countably infinite. The proof is simple: every fraction consists of two integers, the numerator and denominator, and the same technique using prime numbers (3 raised to the numerator multiplied by 5 raised to the denominator) gives a one-to-one correspondence with the natural numbers.
But Cantor proved that the real numbers are uncountable using his famous diagonal argument: given any list of real numbers, you can construct a number Z whose nth decimal digit differs from the nth digit of the nth number on the list, so Z is not on the list. This diagonal method proved exceptionally fruitful and led to many later results, from Russell's paradox to the halting problem.
Hamkins gives the argument a human form: for any collection of people, you can form more committees than there are people, even if there are infinitely many people. If this were not true, you could name each committee after a person, and then form committee D from everyone who does not belong to the committee bearing their name—leading to a contradiction when you ask whether Daniela belongs to the committee bearing her name.
The same logical structure lies behind Russell's paradox, which Hamkins prefers to call Russell's theorem: there is no set of all sets.
The story is dramatic: Frege had almost completed his monumental work on reducing all mathematics to logic when Russell wrote to tell him that his system was contradictory. Frege had to add an appendix to his work in which he wrote that “hardly anything more unwelcome can happen to a scientific author than to have the foundations of his edifice shaken when the work is complete.”
Set theory emerged as a foundation for mathematics through this crisis. The ZFC axiomatic system (Zermelo–Fraenkel with the axiom of choice) was formulated by Zermelo in 1908 in response to challenges to his proof. The axiom of choice says that for every collection of nonempty sets, you can select exactly one element from each. Hamkins gives it a human form through Russell's story about a wealthy man with infinitely many pairs of shoes and socks: for the shoes, you can always choose the left one, but for the socks, which are indistinguishable from each other, there is no selection rule.
Hilbert's program aimed to prove that mathematics is consistent and complete. But Gödel's incompleteness theorems decisively destroyed it: no computably axiomatizable theory that includes elementary arithmetic can prove all true statements, nor can it prove its own consistency. As Hamkins puts it, if Hilbert had been right, mathematics would simply be a mechanical process of enumerating theorems—something that ultimately is not the case.
Hamkins considers the distinction between truth and proof the most beautiful idea in philosophy. Truth concerns what actually holds in mathematical reality, while proof concerns how we interact with it and acquire knowledge. Tarski defined truth through the disquotational theory of truth: the sentence “snow is white” is true if and only if snow is white. This made it possible to define truth mathematically within a structure.
The halting problem shows that there can be no computational procedure that decides, for every program, whether it will halt. The proof is another diagonal argument: we assume such a procedure exists, construct a program Q that behaves in the opposite way to each program P when P is applied to itself, and reach a contradiction when we ask what Q does to Q. This leads directly to Gödel's theorems: if a complete theory of mathematics existed, we could solve the halting problem by waiting for the theorem enumerator to output either “Yes, it halts” or “No, it does not halt.”
But Hamkins and his collaborators proved that the halting problem has a “black hole”: you can correctly decide almost every instance of it, with the proportion converging to 100% as the number of states grows. The reason is almost absurdly simple: in a Turing machine, if the head moves left at the left edge of the tape, it “falls off” and the computation stops. Almost every random program leads to this behavior before repeating a state, so you can answer “does not halt” almost every time.
The continuum hypothesis, Hilbert's first problem, asks whether there is an infinity between the natural numbers and the real numbers. Cantor devoted his life to it without success. In 1938, Gödel proved that the continuum hypothesis is consistent with ZFC by constructing the constructible universe L, and in 1963, Paul Cohen invented the method of forcing and proved that ZFC is also consistent with its negation. The continuum hypothesis is therefore independent of ZFC.
Hamkins draws a more radical philosophical position from this: the set-theoretic multiverse view. Rather than believing that there is a single universe of sets with one truth, he argues that there are many alternative universes with different truths, and that the continuum hypothesis can be switched “on” or “off” depending on the universe you are in. The method of forcing is like traveling from one universe to a larger one, and the “set-theoretic geology” he developed with his collaborators explores whether you can also move backward.
Hamkins also discussed John Conway's surreal numbers, a system that unifies all number systems—natural numbers, integers, rational numbers, real numbers, ordinals and infinitesimals. They emerge from a single process: at each stage, you divide the existing numbers into right and left sets and create the number that fits in the gap between them. Zero is born on the first day, and the process continues transfinitely until “day omega,” when all real numbers are born, along with omega itself and infinitesimals.
On artificial intelligence, Hamkins says he is skeptical about using large language models in mathematical research: he finds that they produce arguments that look like proofs but are not. As he puts it, they are designed to produce something that resembles a logically sound argument, rather than a logically sound argument. He acknowledges, however, that other prominent mathematicians find these systems useful, and that connecting them to verification systems such as Lean is a different matter.
His own working style is social: he has almost a hundred collaborators, and many of his papers grew out of discussions on MathOverflow.
For Hamkins, the most beautiful idea in mathematics is the transfinite ordinal numbers: the idea that you can keep counting beyond infinity—omega, omega plus one, omega plus omega, omega squared, and so on without end.
In philosophy, he does not need to choose whether he is a mathematician or a philosopher: his work unites both fields, and he considers himself to live entirely in the Platonic realm. As he says, physical reality is deeply mysterious, while mathematical existence is something we understand much more satisfactorily.
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