Some Infinities Are Larger Than Others

Lex Fridman opens with Georg Cantor's late-nineteenth-century claim that some infinities are larger than others — a result that, in his telling, broke mathematics before rebuilding it. The discovery set off a theological crisis because infinity had been associated with God, a mathematical civil war in which Leopold Kronecker called Cantor a "corrupter of youth" and tried to block his career, and a personal collapse: Cantor spent his last years in and out of sanatoriums, obsessed with proving the continuum hypothesis. Hamkins wants to start earlier than Cantor. For Aristotle, infinity was potential rather than actual; Archimedes exhausted areas with more and more triangles; for centuries almost every mathematician treated a completed infinite collection as incoherent.

Galileo was the loud exception. In *Two New Sciences* he noticed that the perfect squares line up one-to-one with the natural numbers — every *n* maps to *n*² — which suggests there are as many squares as numbers, even though the squares leave gaps. The same matching works for line segments of different lengths and for concentric circles: a fan of rays pairs every point on the shorter object with a unique point on the longer one. Galileo took the tension as a sign that comparing infinite quantities was itself incoherent, and he threw up his hands.

The contemporary resolution, Hamkins says, is to keep what he calls the Cantor-Hume principle: two collections have the same size if and only if there is a one-to-one correspondence between them. That principle collides with Euclid's principle that the whole is always greater than the part, a rule Euclid used throughout the *Elements* when calculating area. Finite collections never force a choice between the two. Infinite ones do, and Cantor is the person who made the choice sharp by exhibiting two infinite sets that cannot be paired at all.

To make countable infinity concrete, Hamkins walks through Hilbert's Hotel: infinitely many rooms numbered 0, 1, 2, and so on, every room occupied. A new guest arrives. The manager asks every occupant to move up one room; room 0 opens; the new guest checks in. The hotel was full, and it is still full, but one more person fits. Adding an element to an infinite set need not make it larger, which is exactly a violation of Euclid's principle.

Twenty new guests are easy: everyone moves up twenty rooms. When Hilbert's bus arrives — infinitely many seats — the current guests double their room numbers and occupy the evens; the bus passengers take the odds. Two countable infinities have been shoved into one. A set is countable, Hamkins says, if it fits in Hilbert's Hotel, and the union of two countable sets is still countable. If you thought there was only one kind of infinity, that fact would not even be surprising.

Hilbert's train is worse: infinitely many cars, each with infinitely many seats. The manager again frees the odd rooms, then sends the passenger in car *C*, seat *S* to room 3^*C* × 5^*S*. The number is always odd, and unique factorization recovers *C* and *S*. The same trick shows that a countable union of countable sets is countable — a result that shocked Hamkins when he first learned it. A zigzag path through the integer lattice gives a more visual proof: give every infinite collection a turn to add one more person to the list, and everyone gets a room.

The rational numbers look bigger because they are dense — between any two fractions sits another — yet they are still only countable, since a fraction is a pair of integers. After so many infinities have collapsed to the same size, it is tempting to think there is only one infinity. Cantor's achievement was to prove otherwise: the real numbers do not fit in the hotel. The infinity of the continuum is strictly larger than the infinity of the counting numbers.

Assume, for contradiction, that the reals can be listed: *R*₁, *R*₂, *R*₃, and so on. Hamkins constructs a number *Z* whose integer part is 0 and whose *n*th decimal digit differs from the *n*th digit of *R*ₙ, never using 0 or 9 so that the 0.999… versus 1.000… double-representation bug cannot sneak in. Then *Z* differs from every number on the list, contradicting the assumption that the list was complete. The reals are uncountable: there is more than one size of infinity.

The argument is often called a proof by construction: given any list, Cantor hands you a specific real that is missing. What mattered even more, Hamkins says, is the *shape*. Look at the *n*th object, build something that differs from it in the *n*th place, and the result sits outside the whole list. That reflexive pattern is the skeleton of Russell's paradox, of the halting problem, of the recursion theorem, and of almost every major result in mathematical logic. Diagonalization did not stay a curiosity about decimal expansions.

Along the way Fridman asks for a favorite number, and Hamkins sidesteps with a joke-proof that every natural number is interesting. If there were a boring number, there would be a smallest uninteresting number — and that property is itself extremely interesting. Zero is the additive identity, one the multiplicative identity, two the first prime; the rest follow. Then they return to Cantor. The infinity crisis, Fridman notes, is what made set theory a foundation, forced the axiomatization that became ZFC, and opened the fields Gödel and Turing would create.

Set theory, Hamkins insists, plays two roles that people constantly mix up. It is its own subject, full of transfinite constructions and well-founded definitions. It also happens to serve as a foundation, because a set lets you treat a collection of things as one abstract thing — a bag of objects. The axioms of Zermelo–Fraenkel set theory with choice, ZFC, are the standard account of that idea: extensionality, empty set, pairing, union, power set, infinity, separation, replacement, regularity, and choice.

The history is backwards from the textbook order. In 1904 Ernst Zermelo proved that the axiom of choice implies the well-order principle: every set can be well-ordered. There was not yet a listed axiom system; Cantor had not worked that way, and neither had Zermelo. The well-order theorem was so controversial — historians say no theorem had ever been argued about so publicly or so vociferously — that Zermelo was pressed to write down the theory in which the argument could be formalized. That 1908 theory became Zermelo set theory.

Bertrand Russell's closet makes the stakes plain. Infinitely many pairs of shoes are easy: always take the left one. Infinitely many pairs of socks, indistinguishable within each pair, give the butler no rule. What is at stake, Hamkins says, is whether a choice function must be defined by a procedure, or whether mathematical reality is rich enough to contain arbitrary choices you cannot name. As a realist he has no trouble asserting that there is a way of choosing even when he cannot tell you which way it is.

Zermelo and others then looked in the papers of the loudest objectors and found that many of them had been using the axiom of choice implicitly. The principle is so natural that you can invoke it by accident. Gödel and Cohen later showed that choice itself will never be the source of inconsistency: if ZFC is inconsistent, ZF already is. The remaining reason to watch for choice, Hamkins says, is constructivity. An existence proof that leans on it may produce objects you cannot exhibit.

Cantor proved something more general than the uncountability of the reals: for any set whatsoever, the power set — the set of all its subsets — is strictly larger. The argument is diagonalization with the real-number clutter stripped away. Associate every element of *X* with a subset; form the set *D* of elements that are not in their associated subset; *D* must be attached to someone, call her Diana; Diana is in *D* if and only if she is not. There are always more subsets than elements.

Hamkins likes to anthropomorphize. For any collection of people, even infinitely many, you can form more committees than there are people: name every committee after a person, then form committee *D* of those not on the committee named after them. A student at Oxford recast it as fruit salads: more salads than fruits, and the diagonal salad named after durian both must and must not contain durian. All four arguments — reals, power sets, committees, salads — are the same logic.

That logic is Russell's paradox. If the class of all sets were a set, you could form the set of sets that are not elements of themselves, and it would be a member of itself if and only if it was not. Hamkins refuses to call it a paradox. When he teaches it, it is Russell's theorem: there is no universal set. It is not confusing anymore, he says, because the limitation has been absorbed into the basic understanding of what a set is. At the time it was devastating.

Gottlob Frege was finishing a monumental attempt to reduce mathematics to logic, and his system implied the general comprehension principle: for any property, you may form the set of objects with that property. Russell's letter arrived as the volume was already at the printers. Frege added an appendix, writing that hardly anything more unwelcome can befall a scientific writer than to have a foundation shaken after the work is finished. Hamkins's own view is that logicism's main goals were fulfilled by set-theoretic foundationalism: if the principles of ZFC are logical in character, mathematics can be founded on logic after all.

After Russell, Burali-Forti, and the other antinomies, David Hilbert still refused to abandon set theory. "No one shall cast us from the paradise that Cantor has created for us," he said, because set theory had unified mathematics: algebra, analysis, topology, and geometry had been living on separate axiom systems, yet proofs freely borrowed across them. The Hilbert program had two aims. Write down a strong theory, probably set theory, strong enough to answer every mathematical question — "We must know, we will know," as he put it in his retirement address. Then prove, in a weak finitary arithmetic about finite strings of symbols, that the strong theory never yields a contradiction.

Formalism was the philosophical move that made the second aim thinkable. The *meaning* of the infinitary sentences might refer to huge uncountable objects, the source of the paradoxes. The *proofs* themselves are finite sequences of symbols that follow finitary rules. Divorce meaning from manipulation, and you can treat mathematics as a game whose safety you might certify without trusting the infinite. If Hilbert had been right, Hamkins says, mathematical research would become turning the crank of a theorem-enumeration machine: every question answered by waiting for yes or no to print. The fundamental nature of mathematics would be rote computation.

Gödel's incompleteness theorems of 1931 take both aims down. The first says you cannot write a computably axiomatizable theory that includes a certain amount of arithmetic and answers every question: some statements will be neither proved nor refuted. The second says no such theory can prove its own consistency — so the weak finitary theory cannot certify the strong one, and the strong one cannot certify itself. Would you trust a theory that proves itself consistent? Hamkins compares it to a used-car salesman telling you he is trustworthy. An inconsistent theory would prove its own consistency too.

If Hilbert is wrong, mathematical reality has two permanent features. You will never write down a theory that settles every question, and you will never have a fully convincing certificate that your working theory is free of contradiction. That, Hamkins says, is exactly the world Gödel revealed. He does not find it traumatic. It is a profound fact about our situation with regard to mathematical truth, and now that we know it we can move on. In logic, a "theory" is any set of sentences in a formal language; you may be able to decide the axioms and still be unable to decide the theorems.

The distinction Hamkins calls core to modern logic was, before Gödel and Alfred Tarski, routinely blurred. Even Bourbaki's graduate text conflated the two: to be true meant to be provable. Truth sits on the semantic side. It is a property of a sentence *in a structure* — a field, a group, a graph — and Tarski's disquotational account is the one Hamkins wants. The sentence "snow is white" is true if and only if snow is white. The quotation marks come off: a claim about a sentence is traded for a claim about the world.

That idea lets you define truth in a formal language by induction. A structure satisfies "φ and ψ" just when it satisfies φ *and* it satisfies ψ — the first *and* is inside the object language, the second is in the metalanguage. The same disquotation handles the other connectives and the quantifiers. A sentence is never true in the abstract; you have to say which structure you mean. Proof, by contrast, is Hilbert's syntactic object: a sequence of sentences that follows allowed rules, such as modus ponens. If you know *A* and you know *A* implies *B*, you may write *B*.

A decent proof system, Hamkins says, needs three properties, not the two logicians usually name. It should be *sound*: if the premises are true in a structure, so is the conclusion. It should be *complete*: whenever a sentence is a logical consequence of some assumptions, there is a proof of it. And it should be *computably decidable* whether a given sequence is a proof at all. A mathematician who claims a proof that will not fit in the margin has not offered a proof. Soundness, completeness, and checkability together are what make the concept usable.

The halting problem asks, given a program, whether it will ever finish. For any one instance the answer is yes or no; the question is whether a single computable procedure answers every instance. There is not. You can get all the yeses by running the program until it stops. The noes require insight you cannot compute uniformly: waiting a thousand years without a halt does not license the claim that it never will. The proof is another diagonal. Assume a subroutine *H* that decides whether *P* halts on *P*. Build *Q* that asks *H* that question, then loops forever if the answer is yes and halts immediately if the answer is no. *Q* on *P* does the opposite of *P* on *P*. What does *Q* do on *Q*?

From undecidability of halting, Gödel's first theorem drops out without a Gödel sentence. Suppose you had a complete computable axiomatization of elementary finite mathematics. Then a theorem-enumeration machine would eventually print either "*P* halts on this input" or "*P* does not." That would solve the halting problem, which is impossible. Hamkins calls this the simplest proof of Gödel he knows. The Entscheidungsproblem — whether a given statement is a consequence of a given theory — is the same undecidable question in other clothes.

Asked about the art of proof, Hamkins describes the undergraduate book *Proof and the Art of Mathematics*, dedicated to his students: may all their theorems be true, proved by elegant arguments that flow from hypothesis to conclusion. His favorite classroom problem asks whether a circle of people can point at one another so that everyone is pointed at more times than they point. Give a dollar to each person you point at. Then everyone would make money, which a closed group cannot do by trading with itself. The Twitter version is the same: you cannot arrange that everyone has more followers than following.

The dollar argument is convincing because it borrows a bodily fact — making money is hard — in place of an abstract lemma about functions. If the group is infinite, the theorem fails. Countably many friends, each with one dollar, can pass bills so that everyone ends up with infinitely many: send everyone on the *n*th Hilbert train car to the person who lands in the *n*th hotel room. Cardinality does not notice. Anthropomorphizing, Hamkins tells students, is not decoration. It is a way of replacing a claim you would have to prove with a claim your life has already taught you.

Does infinity exist? Hamkins treats the question as no different from asking whether the number five exists. People who find abstract objects suspicious often want to reduce mathematical existence to tables and chairs. He thinks the impulse is backward. Describe a steam locomotive in every engineering detail, then ask what it would mean for that locomotive to exist physically rather than imaginarily. The only answer available is "it exists in the physical world," which restates the question. Physics has made the mystery worse, not better: billiard-ball matter gave way to atoms, then quarks, then wave functions in a probability cloud.

The empty set, by contrast, he can describe. The singleton of the empty set, too. Deeper in, the account gets clearer rather than murkier. He lives, he says, in the Platonic realm; he does not really understand the physical universe. We have rich *experience* of tables, birthdays, and martinis, but experience is not an account of what physical existence is. On this he is a realist: abstract objects have a real existence we can actually talk about. Fridman presses on what is more real, the world we see or the world of theorems. Hamkins will not say more real. He will say better understood.

Structuralism is the attitude that fits this ontology. What matters about a mathematical object is not its substance but how it functions in a structure, and structures are considered only up to isomorphism. The number four could be played by a bottle of water; the copy would be mathematically identical. Frege's Julius Caesar problem asked, on the Cantor-Hume principle, how we know Caesar is not a number. Structuralists shrug. Give me a number system; if Caesar is not in it, swap him in for 17. The question of essence is not a mathematical question. That, Hamkins adds, is why modern set theory dropped ur-elements: any structure with atoms is isomorphic to one without them.

A thousand years from now, will mathematics still have upheavals like the early twentieth century? Hamkins has a foot in both mathematics and philosophy, and the cultures of progress differ. Mathematics keeps improving its understanding until the old questions are no longer the questions; he sees no reason for that to stop, and the mathematics of a thousand years hence would probably be unrecognizable to him. Philosophy still carries eternal questions that many philosophers treat as more valuable to pose than to answer. If Archimedes could communicate, Hamkins thinks he could still teach him some of what we now know — a bet on simple arguments remaining transmissible. MathOverflow, where he has been the highest-rated user since shortly after it launched in 2009, is where a lot of that teaching happens in public: logic-adjacent questions in group theory and analysis forced him to learn neighboring fields in order to answer.

Once you know the reals are strictly larger than the naturals, the next question is almost forced: is anything in between? The continuum hypothesis says no. Cantor spent his life on it. Every set of reals anyone could point to was either countable or equinumerous with the whole line. He had a program: prove CH for open sets (easy — any interval matches the line via arctangent), then closed sets (the Cantor–Bendixson theorem, for which he had to invent the ordinals), then Borel sets, then projective sets. The climb has been carried astonishingly far, and at the projective level it needs large-cardinal assumptions. It cannot finish. The hierarchy of complexity does not exhaust all sets of reals, and independence blocks the rest.

Hilbert put CH first on his 1900 list of 23 problems, less as a prophecy of the century, Hamkins suspects, than as a list of things he found fundamental. Set theory was already supplying the unity of mathematics. In 1938 Gödel built the constructible universe *L* and showed that if ZF is consistent, so is ZFC plus CH: you cannot refute the continuum hypothesis. In 1963 Paul Cohen invented forcing and showed that if there is a model of set theory, there is one where CH is false: you cannot prove it either. Hilbert's first problem, in the standard axioms, has no answer. Forcing, as Hamkins thinks of it, is a method for traveling between mathematical universes.

Independence is not a rarity. Practically every nontrivial statement of infinite combinatorics is independent of ZFC. A Berkeley analyst whose question turned out independent said, "I guess I asked the wrong question." Set theorists celebrated: the question had carved nature at its joints. The universe view, or monism, answers by saying ZFC is merely weak and we should hunt for the one true set theory, perhaps via large cardinals. Those axioms give an endlessly tall consistency-strength hierarchy, exactly as Gödel's second theorem predicted — and none of the known large cardinals settle CH. Hamkins takes the last half-century of constructions as evidence that there is no unique set-theoretic reality. Given any model, you can force CH on or off like a light switch. That plural fact *is* the answer.

The philosophical split does not change the theorems. It changes which questions you find worth asking. Hamkins's advisor Hugh Woodin, a leading universe-view figure, chases the nature of one true universe through ultimate *L*. Hamkins's pluralism led him to the modal logic of forcing with Benedikt Löwe, to set-theoretic potentialism, and with Gunter Fuchs and his student Jonas Reitz to set-theoretic geology: start from a universe and ask which smaller ground it might have been forced from. Reitz had insisted on going backwards; Hamkins first said forcing does not work that way, then took the idea seriously. Geology grew from a pluralist picture and was later adopted by universe-view researchers studying the mantle of the one true cosmos. You can even prove something in a forcing extension and pull the conclusion back to the ground model, the way sixteenth-century algebraists marched through square roots of negatives and came home with a real answer.

John Conway's surreal numbers, Hamkins says, unify the integers, the reals, the ordinals, and the infinitesimals in a proper class generated from nothing by one rule. At each stage, split the numbers you have into a left set and a right set with everything on the left less than everything on the right, and create the number that fills the gap. Day zero is the big bang of numbers: two empty sets birth zero. Then come 1 and −1, then 2 and ½, and on day ω every real is born, along with ω itself and an infinitesimal ε. The surreals form a real closed field — you can add, multiply, divide, take square roots, and every odd-degree polynomial has a root — but they have no least-upper-bound property and no convergent sequences. They are, he says, fundamentally discontinuous. Calculus on them has to go through infinitesimals, not limits.

At a New York conference someone asked Conway, rudely, what his greatest disappointment in life was. Conway answered with the *reception* of the surreals: he had wanted them to become a fundamental tool across mathematics and science, and they never quite did. Philip Ehrlich told Hamkins that Conway was his own worst enemy, because he treated everything as a game and the surreals as a toy. Hamkins's view is the opposite. They are serious, useful, and profound; he has been writing a series on them at *Infinitely More*. Conway's Game of Life, Fridman adds, is another playground of that style — and Hamkins notes that whether a given cell ever becomes alive is equivalent to the halting problem.

Alexey Myasnikov once asked whether the halting problem has a "black hole": a tiny hard region outside which almost every instance is easy. With the standard one-way-infinite Turing-machine model, they proved it does. There is an easily recognized class of programs for which halting is easy to decide, and as the number of states grows the class's density goes to 100%. About 13.5 percent never halt for the trivial reason that they have no transition into the halt state. The jackpot reason is stupider: a random machine's head does a one-dimensional random walk from the leftmost cell, and by Pólya's recurrence theorem it almost surely returns and then falls off the tape before repeating a state. Undecidability is highly local. The same pattern, Hamkins says, shows up in complexity: most NP-complete problems have polynomial-time approximations that solve almost every instance, which is why SAT solvers work so well in industry.

P versus NP, he warns, is about asymptotic behavior; the polynomial could have enormous degree and coefficients, so slogans about civilizational wealth if P = NP should be discounted. We already have an algorithm, just a terrible one, and practical approximations already solve the engineering cases. Forced to name the greatest mathematician, he would pick Archimedes for achievements that transcended their era, but he does not rank people and wants insight from whoever can provide it. His own style is a dislike of difficult mathematics: simple, clear arguments for surprising theorems, preferably ones he can hold entirely in his head. The method is playful curiosity — tweak a definition, try a favorite example, press on one side — plus anthropomorphizing models as places you live in and forcing as travel. Andrew Wiles's seven-year isolation is the opposite of his practice. He is pushing toward a hundred collaborators; many papers grew from MathOverflow threads. He could never do what Wiles did, and he tends to think he would not solve the problem by locking himself in a bedroom either.

Infinite chess started as a MathOverflow question from number theorist Noam Elkies: what is a chess piece, mathematically? The board is the integer lattice, unbounded in four directions; pieces move as usual, with no pawn promotion and with infinite play counted as a draw rather than threefold repetition. Hamkins constructed positions that White will definitely win, though there is no finite *N* such that White can force mate in *N* moves. Black controls how long it takes. The game value is the ordinal ω: Black must name a finite number on the first count, then subtract one each time. With US national master Corey Evans — also a law-and-philosophy professor, and once Hamkins's son's chess coach — and later PhD student Norman Perlmutter, they reached ω⁴. It is now known that every countable ordinal arises as the game value of some position. The constructions were not algorithmic. Hamkins designed the layouts; Evans kept finding hanging pawns and bishops leaking out of cages.

On current large language models as mathematical collaborators, Hamkins is blunt: he has tried paid systems and found them basically of zero use, typically returning garbage, then insisting the broken step is "totally fine." He would stop talking to a person who did that. He is surprised that mathematicians he respects say they find the tools helpful, and he wonders whether the ideas are still coming from the humans. His deeper objection is about method. The model is trying to give him an argument that *sounds like* a proof rather than an argument that *is* a proof. As an undergraduate at Caltech he typeset homework in the then-new LaTeX so beautifully that it looked like a published book, and he stopped being critical enough: the grades came back terrible. Chatbots amplify that effect. Wiring a model to Lean is a different game, because the checker does not care how a proof looks. An ordinary person asking a chatbot for a mathematical argument, he thinks, is a dangerous source of error.

Fridman counters that collaborating with an LLM is a real skill: you have to load in your work and your way of thinking, then treat the system as a source of connections rather than answers, with extreme patience for how often it is dense. In programming, where the training data is vast, he already sees a path. Hamkins grants the point and adds a dry one of his own: MathOverflow answers are surely in the training data, so in a sense he would be talking to himself. Asked for the most beautiful idea in mathematics, he names the transfinite ordinals — Cantor's system for counting past infinity. After 0, 1, 2, … comes ω, then ω+1, then ω·2, then ω², a limit of limits, and it never stops. That simple idea underwrites Cantor–Bendixson, von Neumann's cumulative hierarchy, Gödel's *L*, and Zermelo's well-order proof. The most beautiful idea in philosophy, he says, is still the distinction between truth and proof: truth is about the way things are, proof about how we come to know them. Fridman adds that the gap between the two is full of mysteries in the Platonic realm, in physics, and in human affairs. That gap is the journey, whatever field you are in.