#488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins

Today I’m talking with Joel David Hamkins, a mathematician and philosopher known for set theory, the nature of infinity, and some mind-bending ideas about truth and paradox. I’ve been laying low, reading and working, and I just want to say how lucky I feel to be here and to be shown so much love—thank you; now let’s dive in.

Some infinities outrun others, which cracked mathematics before rebuilding it—religious shock, academic fights, paradoxes like Russell’s, and Cantor’s own struggles; can you explain the idea and why it transformed the field?

I’d start long before Cantor: Aristotle favored potential over completed infinity, Archimedes used exhaustion, and Galileo noticed one-to-one matches that clash with “the whole is greater than the part.” That tension—between size by bijection and Euclid’s intuition—set the stage for Cantor’s clarity.

Hilbert’s Hotel makes it vivid: a fully booked hotel with rooms labeled by natural numbers still fits one more guest by shifting everyone up a room. Infinity breaks the old rule that adding an element must make a set larger.

So the hotel is full, yet you can still tuck in a newcomer—an infinity-only phenomenon.

You can also take twenty guests at once by shifting up twenty rooms, or even a bus of countably many guests by moving current occupants to even rooms and placing newcomers in the odds.

That’s packing one infinity neatly inside another.

We call a set countable if it can be listed like the natural numbers, and unions of two countably infinite sets are still countable. Even a train with infinitely many cars, each with infinitely many seats, fits by coding car and seat as a single odd number using prime powers.

There’s also a clean visual: arrange pairs on a grid and zigzag through the lattice, guaranteeing everything appears on one list.

Great; now what makes the real numbers a larger kind of infinity?

First, rationals are still countable, despite being dense, because each fraction is just a pair of integers. Cantor’s big leap is that the reals can’t be listed—there are simply more of them.

Reals include rationals and irrationals; rationals are just ratios of integers.

Among reals, algebraic numbers solve polynomial equations, while transcendental numbers don’t; Liouville showed they exist, and Cantor’s work implies almost all reals are transcendental.

Transcendentals aren’t roots of nonzero integer-coefficient polynomials—think beyond algebraic formulas.

Classics like pi and e are transcendental, and by the way, there’s a cute argument that every natural number is “interesting” because the smallest “boring” one would be interesting for that very reason.

Alright, back to Cantor’s diagonal.

Assume you’ve listed the reals; build a new number by changing the nth digit after the decimal to differ from the nth number’s nth digit, avoiding zeros and nines to dodge dual representations. This diagonal construction can’t be on the list, which proves the reals are uncountable and introduces a method that powers much of modern logic.

This crisis led to a rebuild: set theory as foundation, axioms like ZFC, and new fields in logic—what is set theory and how does it ground math?

Set theory is both its own subject and a foundation; it lets us treat many things as one thing, a set, and reason with axioms about membership and construction.

So we assume clear starting facts about sets and build math on top of them.

Historically, Zermelo shaped the axioms while defending the axiom of choice and the well-ordering theorem, formalizing what Cantor did informally and sparking decades of debate.

The axiom of choice says you can pick one element from each nonempty set, even without a rule; controversial then because there wasn’t yet an axiomatic language.

Choice feels natural—like a function that selects one from each set—but when no rule exists we still assert existence; think shoes versus socks, where indistinguishable pairs force you to rely on choice. Constructivists push back because a proof that only claims existence without a method can feel unsatisfying.

Can you give the flavor of ZFC’s axioms?

Extensionality says sets are determined only by their members; others guarantee empty sets, pairing, unions, power sets, and an infinite set. With these, Zermelo showed every set can be well-ordered.

It’s wild to imagine formalizing all that as it was happening.

Debate was fierce—many unknowingly used choice while opposing it—and later results by Gödel and Cohen showed choice isn’t the source of inconsistency, though constructivist concerns remain.

And consistency just means no contradictions can be derived; now, could you give the diagonal take on uncountability and Russell’s paradox?

Cantor’s theorem says the power set of any set is strictly larger by forming the “not-in-your-own-set” diagonal subset, which can’t match any single element. Recast as committees or fruit salads, the same logic also shows there’s no universal set, the insight Russell used to upend Frege’s system with a single devastating contradiction.

That must have been crushing for Frege—clean, simple, and fatal.

Logicism didn’t die, but I think ZFC fulfills its core aims if you view these axioms as fundamentally logical, though some disagree about infinity and choice.

Let’s go to Gödel’s incompleteness theorems—what do they say about mathematical truth?

Hilbert wanted a powerful theory to answer every question and a finitary proof that its reasoning is safe; formalism helps by treating proofs as symbol manipulation. That vision is attractive, but Gödel showed it can’t succeed.

So proofs become tools operating on symbols inside the system.

Exactly; a finitary base like Peano arithmetic can formalize most classical number theory, even if people argue how “finitary” it must be. It lets us reason about proofs themselves while keeping statements and their meanings separate.

And Hilbert’s program is to formalize everything and prove those systems can’t contradict themselves.

If he’d been right, math would reduce to churning out theorems by machine; instead, Gödel proved no computably axiomatized theory encompassing basic arithmetic can answer every statement, and no such theory proves its own consistency.

When you say theory, you mean an axiomatic system and its consequences; deciding theorems differs from enumerating them.

Right; axioms can be listed computably, but the set of all consequences isn’t decidable—only enumerable by running through all proofs.

That ties to the halting problem—what is it?

Given a program, will it ever stop; you can recognize the yes-cases by waiting, but no single procedure answers all instances. A diagonal program that flips behavior on its own input yields a contradiction, and if arithmetic were complete you could solve halting by waiting for the machine to prove either outcome, so incompleteness follows.

You’ve also written about the art of proof—what makes a good one?

I love teaching proofs through lively problems; for example, in a circle of friends, it’s impossible for everyone to be pointed at by more people than they point to—if each pays a dollar to everyone they point at, the group can’t all profit just by shifting money. Anthropomorphizing helps the structure snap into focus.

Proofs can make the nonobvious feel inevitable.

With infinitely many people, the story flips; by Hilbert-style accounting, everyone can end up with infinitely many dollars with only finite giving by each person.

So is infinity real in the same way numbers are?

Questions about infinity mirror questions about the number five; in fact, physical existence grows murkier the more physics reveals, while abstract existence in set theory is, if anything, clearer and more transparent to analyze.

If numerals are just representations, do numbers themselves exist, and how should we think about their reality?

I'm a mathematical realist, but I also lean structuralist: what matters are roles in a structure, not the substance, so any isomorphic copy is mathematically equivalent.

So numbers are like pointers into structure rather than things with essence?

Yes, the interest is in how objects behave together; asking what four “really is” is beside the point, which is why the Julius Caesar puzzle misses the structural view.

You’ve suggested math may be more graspable than the physical world; which is more real to you?

I feel at home in a platonic mathematical realm; physical existence remains deeply mysterious, while abstract existence is something we can analyze more convincingly.

Do you expect breakthroughs over the next thousand years, and how does mathematical progress compare to philosophy?

Math shows relentless progress and reframes its own questions; philosophy advances too, but many core questions persist and often emphasize asking over answering.

You’re a legend on MathOverflow; how do you choose and solve questions there?

I chase what intrigues me, often logic or logic-adjacent problems; I learn just enough of neighboring fields to bridge set theory and their context, which has helped me grow a lot.

Let’s set up the continuum hypothesis and Cantor’s quest.

Cantor showed the reals are a larger infinity than the naturals, then asked if there’s a size in between; for all definable sets people could analyze, every case collapsed to either countable or the full continuum.

Can you outline why open and closed sets don’t give a counterexample?

Nontrivial open sets match the whole line, and Cantor–Bendixson proved every closed set is either countable or the size of the reals; this extends through the Borel hierarchy, and with large cardinals through projective sets.

But ultimately CH is independent of ZFC; what is the story there?

Gödel built the constructible universe where choice and CH hold assuming consistency; Cohen later invented forcing to build models where CH fails, so CH can’t be proved or refuted from ZFC.

Forcing feels like moving between mathematical worlds, which ties to your multiverse stance.

Exactly; thousands of independence results show rich alternative universes, so I see plural truth across a multiverse, not one final set-theoretic reality.

Is independence a dead end, or can it be fruitful?

It’s fruitful; independence reveals genuine splits in mathematical reality, and large cardinal axioms give a towering hierarchy of stronger theories, even though they still don’t settle CH.

What does your multiverse view enable in practice?

It redirects attention to relationships among universes—forcing, potentialism, and “geology” that studies grounds and mantles—work that even universe-view researchers now use.

Can you prove something in an extension and pull it back down?

Often yes: argue in a forcing extension to see a consequence already true in the ground model, much like early algebraists manipulated complex terms and ended with real solutions.

Switching gears, what are surreal numbers and how are they built?

Conway’s surreals unify naturals, integers, rationals, reals, ordinals, and infinitesimals by a simple generative rule: at each stage, split earlier numbers into left and right sets with a gap, then create the number that fits that gap.

Give the quick growth picture and key properties.

Zero is born from the empty split, then ±1, then diadic rationals, then at day omega the reals, an infinite ordinal, and infinitesimals; with recursive operations they form an ordered real-closed field, yet lack least upper bounds and have no convergent sequences, so calculus works there via infinitesimals, not limits.

Conway’s Game of Life also shows deep complexity.

Life is a playground for undecidability; whether a cell ever turns on is equivalent to the halting problem, so many behaviors are provably beyond complete prediction.

Can we say anything useful about typical programs and halting?

Yes; the halting problem has a “black hole” phenomenon with almost-everywhere decidability—most random Turing machines crash before repeating a state, essentially by a one-dimensional random walk that falls off the tape, so asymptotically you can answer correctly on a density-one set.

How do you see P versus NP and practical impact?

It’s an asymptotic question, so even P equals NP could yield absurdly impractical algorithms; in practice, heuristic SAT and NP-complete solvers already crack almost all real-world instances efficiently.

Greatest mathematician ever?

I don’t rank people, but if pushed I’d say Archimedes; more broadly, discoveries often arrive in parallel because ideas are in the air.

What’s your creative process?

I prefer simple, transparent proofs that surprise; I’m wary of overly intricate arguments, and I work by playful curiosity, constant small variations, vivid thought experiments, and even game-like metaphors to see strategy.

And on working style—Wiles-like solitude or Tau-like switching?

I lean social and collaborative; MathOverflow and joint work have sparked many of my papers, which suits my style far better than long solitary marathons.

Working alone in math can feel brutal, with false dawns and long stretches without a partner to bounce ideas off. In that spirit, what do you make of Grigori Perelman turning down top prizes and saying recognition is irrelevant if the proof stands?

Math has room for many temperaments, and the ideas matter most. Perelman solved a towering problem and chose to live by his own values, which I respect.

Like Olympic athletes, the greats often chase the craft for its own sake. Money and fame may help, but the pull is the art.

I feel the same; the questions drive me. Awards are nice, but they are not the point.

And your MathOverflow work is clearly not for wealth or power, just curiosity.

If we insisted on an objective metric for “greatest,” we could jokingly use MathOverflow scores. I even asked my daughter’s boyfriend for his chess rating and his MathOverflow score.

Speaking of chess, what is infinite chess, and how do you formalize it?

It is standard chess played on an integer grid that extends without edges, with the usual piece moves but no threefold repetition rule, no promotion, and draws defined by infinite play. You can build positions where White will win in finitely many moves, yet for every specific number N it is not mate in N, because Black can stretch the game length by effectively choosing a countdown tied to the ordinal omega.

I love that your interest started from a MathOverflow question; can those positions be constructed algorithmically?

It took creative design with my co-author Cory Evans, a strong player and philosopher, who kept catching tactical leaks as we iterated. We built positions of increasing ordinal game value—omega, then omega squared, omega cubed, and later with Norman Perlmutter we reached omega to the fourth.

Right, the paper on a position with game value omega to the fourth.

Since then, the field advanced to show that every countable ordinal appears as a game value in infinite chess.

Do you see AI becoming a real collaborator for mathematicians?

So far I find large language models unreliable for proofs; they produce arguments that sound right rather than are right, which is dangerous without formal checking. I learned long ago that beautiful typesetting can disguise errors, and these systems risk the same illusion unless paired with verification tools.

With careful prompting and lots of context, I find them helpful for inspiration and cross-linking ideas, especially in programming where training data is rich. They are not sages, but they can be patient companions if you are patient too.

I suspect MathOverflow answers are in the training mix, which makes this all the more interesting.

What is the most beautiful idea in mathematics to you?

Transfinite ordinals, the leap to counting beyond infinity—omega, omega plus n, limits like omega squared—and the transfinite recursion they enable. That simple step underpins set-theoretic hierarchies, constructibility, and well-ordering arguments with the axiom of choice.

And in philosophy, what idea moves you most? Also, thank you for this conversation.

I live between math and philosophy and do not need to choose. The deepest idea for me is the distinction between truth and proof—truth as objective mathematical reality, proof as our route to knowing it—which shapes how I see logic and knowledge.

That gap is where the mysteries live, in math and in human affairs. I am grateful for your mind and your warmth; it was a joy talking with you.

Thank you; it was a real pleasure.

Thanks for listening to my conversation with Joel David Hamkins.