Lex Fridman Podcast #488: Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse

Guest: Joel David Hamkins
Recorded: December 31, 2025

Overview

In this episode, Lex Fridman sits down with Joel David Hamkins—a mathematician and philosopher famed for his work in set theory and the foundations of mathematics—to tackle some of the most mind-bending ideas in math and logic. Topics span from the nature and paradoxes of infinity to Gödel’s incompleteness theorem, the continuum hypothesis, set theory as the foundation of mathematics, and the multiverse view of mathematical reality. The discussion is wide-ranging, deeply technical but highly engaging, offering insights into the philosophy behind mathematics, the personal histories of its creators, and the current state of foundational research.

Key Discussion Points & Insights

1. Infinity: History, Paradoxes, and Cantor’s Revolution

• Historical Attitudes:

  • For thousands of years, infinity was treated as a “potential” concept; the idea of actual, completed infinity was avoided by mathematicians from Aristotle to Archimedes.
  • Galileo, as a rare exception, argued with the orthodox view, anticipating aspects of Cantor's later work but ultimately finding the notion incoherent.
    • Galileo’s Paradox: Paradoxes arise when comparing the set of all natural numbers to the set of perfect squares; both can be placed in one-to-one correspondence, yet there are “gaps” between squares (17:50).

• Cantor’s Breakthrough:

  • Cantor formalized the idea that some infinities are larger than others, “breaking” traditional math (23:00).
  • Developed the Cantor-Hume Principle: sets have the same size if they can be put in one-to-one correspondence—even infinite sets.
  • Introduced the concept of countable and uncountable infinities, with the real numbers being uncountable.

• Hilbert’s Hotel:

  • A classic illustration showing the strange properties of infinite sets. Even if every room is full, new guests (even infinite busloads) can still be accommodated by shifting existing guests (23:04–27:05).
  • Notable quote (Hamkins): “It’s a property of infinity that sometimes when you add an element to a set, it doesn’t get larger.” (25:22)

• Uncountable Infinity:

  • Cantor’s diagonal argument proves that the real numbers outsize the set of natural numbers and rational numbers.

2. Foundations: Set Theory and the Axioms of Mathematics

• Set Theory’s Dual Role:

  • As both a branch of mathematics and the foundation for all of math (48:05).
  • Central insight: a set is an abstract “bag” containing elements, and set theory’s axioms define how these bags behave.

• ZFC (Zermelo-Fraenkel with Choice):

  • The standard axiomatic system for modern mathematics—provides a rigorous foundation (53:04).
  • Axiom of Choice: “For any collection of non-empty sets, it is possible to select exactly one element from each set, even if no explicit rule to make the choice is given.” (51:03)
    • Russell’s “Shoes and Socks” analogy: picking left or right shoes is straightforward, but indistinguishable socks require the axiom (53:12).

• Russell’s Paradox & The Fruit Salad Committee:

  • Russell’s Paradox shows the problem of forming “the set of all sets that do not contain themselves.”
  • Hamkins offers memorable analogies involving committees and fruit salads, highlighting the underlying logic: With any collection, there are more possible subsets (committees or salads) than there are elements (people or fruits) (66:57).
  • Notable quote (Hamkins): “For any collection of people, you can form more committees from them than there are people, even if there are infinitely many people.” (66:57)

3. Gödel’s Incompleteness Theorems & the Nature of Mathematical Truth

• Context and Hilbert’s Program:

  • Hilbert wanted to formalize all of mathematics into an axiomatic system and prove its consistency using only finitary means (77:22).
  • Tension between “proof” (syntactic, mechanical manipulation) and “truth” (semantic, about reality).

• Gödel’s Results:

  • First Incompleteness Theorem: Any consistent, computably axiomatizable system strong enough for basic arithmetic will be incomplete; there are true statements it cannot prove.
  • Second Incompleteness Theorem: No such system can prove its own consistency (85:47).
  • Notable quote: “...the incompleteness theorem should be viewed as a decisive refutation of the Hilbert program. It defeats both of those goals decisively, completely.” (85:47)

• Truth vs. Proof:

  • Truth pertains to what is semantically or model-theoretically the case; proof pertains to what can be derived formally via rules (93:44–97:41).
  • Notable quote: “Truth is about mathematical reality; proof is about our knowledge or interaction with that reality.” (241:15)

• Halting Problem and Diagonal Arguments:

  • The undecidability of the halting problem (no algorithm can decide for every program whether it halts) is connected to Gödel’s theorem (via diagonalization), Turing, and others (104:53–110:00).

4. The Continuum Hypothesis and the Multiverse

• Continuum Hypothesis (CH):

  • Cantor asked whether there is an infinity strictly between that of the integers and the real numbers (142:38).
  • Became the first of Hilbert’s 23 problems.

• Independence from ZFC:

  • Gödel showed (1938) that CH can’t be refuted from ZFC if ZFC is consistent (true in his constructible universe, "L") (157:54).
  • Cohen (1963) invented forcing and showed that CH can also not be proved from ZFC—i.e., it’s independent (158:31).
  • Notable quote: “...the continuum hypothesis is independent of all known large cardinal axioms. So none of the large cardinal axioms...can settle the continuum hypothesis.” (167:58)

• Forcing & The Set-Theoretic Multiverse:

  • Forcing is a powerful tool to “travel” between different models of set theory, each with potentially different truths (161:20).
  • Universe vs. Multiverse View:
    • Universe View: There is a unique, true set-theoretic universe where every question has a definite answer.
    • Multiverse/Pluralist View (Hamkins's position): There are many legitimate set-theoretic “worlds” or universes, and some truths are plural by nature (168:09–170:17).
    • Notable quote: “...the fundamental nature of set-theoretic truth has this plural character....” (170:33)

5. Surreal Numbers and Infinite Chess

• Surreal Numbers:

  • A staggering number system unifying real numbers, ordinals, infinitesimals, and more, invented by John Conway (180:42).
  • Numbers are generated by recursively dividing all previously constructed numbers into left and right sets and forming new numbers in the "gap."
  • The system is not a set but a proper class.
  • Notable quote: “The surreal numbers...extend the integers, rationals, reals, ordinals, and infinitesimals...all sitting inside this colossal system of numbers.” (180:55)

• Infinite Chess:

  • Chess, but played on the infinite integer grid, focusing on bespoke positions with interesting properties (220:21).
  • Collaboration and mathematical creativity are key for constructing positions with high ordinal values—positions where “White can win, but Black controls how long it takes.”
  • Notable quote: “There are positions in infinite chess that White can definitely win in finitely many moves...but there’s no particular N for which White can guarantee to win in N moves.” (222:33)
  • The study inspired and born out of discussions on Math Overflow.

6. Philosophy of Mathematics: Structures, Reality, and the Platonic Realm

• Structuralism in Mathematics:

  • Structuralism: What matters is not what mathematical objects are made of, but their relationships within structures (126:47).
  • The question “Is Julius Caesar a number?” illustrates the irrelevance of “essence”—one can replace an element with another and still have a valid structure.
  • Notable quote: “...the substance of individuals in a mathematical structure is irrelevant for any mathematical property of that structure.” (129:05)

• Mathematical vs. Physical Reality:

  • Hamkins argues the existence of numbers or infinity is, in some sense, more “real” or accessible than physical existence, which physics keeps making more mysterious (120:46).
  • Notable quote: “We don’t really have any understanding of what the physical world is as opposed to the abstract world. And it’s the abstract world where existence is much more clear.” (125:56)

7. Math Overflow, Collaboration, and the Art of Mathematical Proof

• Math Overflow Legend:

  • Hamkins has answered thousands of research-level questions, focusing not just on his core specialty but learning adjacent subjects to provide insights (138:46–141:23).
  • His collaborative, play-centered, curiosity-driven approach to mathematics stands in contrast to solitary grinders like Andrew Wiles, whose achievement he still admires (213:54).

• Proof as Art and Science:

  • Effective proof writing shouldn’t be mechanical; it should display compelling, beautiful ideas (112:45).
  • Notable quote (book dedication): “May all their theorems be true, proved by elegant arguments that flow effortlessly from hypothesis to conclusion while revealing fantastical mathematical beauty.” (114:35)

• Anthropomorphism as Mathematical Tool:

  • Hamkins encourages visual and narrative thinking—imagining mathematical objects as people or things with agency to gain intuition and discover proof strategies (116:32–119:11).

8. Artificial Intelligence and Mathematics

• Skepticism about AI as a Mathematical Collaborator:
  • Hamkins has found current LLMs unhelpful or even dangerous as they can generate plausible-sounding but incorrect mathematics (229:14–234:02).
  • Recognizes, however, that collaboration or inspiration may become possible as the systems improve.

9. Favorite Ideas and Final Reflections

• Most Beautiful Math Idea: The transfinite ordinals and the possibility of “counting beyond infinity” (238:32).
• Most Beautiful Philosophy Idea: The deep, essential distinction between truth and proof (240:57).
• On Progress: Mathematics is unique in its steady, cumulative progress, unlike philosophy where the great questions linger eternally (137:42).

Memorable Quotes and Moments

• On Infinity:
  “Some infinities are bigger than others.” (Lex, 16:58)
• On the Axiom of Choice:
  “When you can describe a specific way of choosing, then you don’t need to appeal to the axiom to know that there’s a choice function…. But the problematic case occurs…infinite collection of socks... the butler wouldn’t have any kind of rule for which sock in each pair to pick.” (Hamkins, 53:04–53:53)
• On Set Theory:
  “Set theory has also happened to serve in this other foundational role…provides a way to think of a collection of things as one thing.” (48:05)
• On Gödel’s Theorem and Hilbert’s Dream:
  “The first incompleteness theorem says you cannot write down a computably axiomatizable theory (which) answers all the questions…every such theory will be incomplete...” (85:47)
• On Independence:
  “When you ask a question that turns out to be independent, then you asked exactly the right question…because you found this cleavage in mathematical reality.” (163:54–165:15)
• On Life in the Platonic Realm:
  “I live entirely in the Platonic realm. And I don’t really understand the physical universe at all.” (132:21)
• On Proof and Truth:
  “Truth is about objective reality, whereas proof is about how we come to know the things that are true about the world.” (241:15)
• On AI in Math:
  “The AI is trying to give me an argument that sounds like a proof rather than an argument that is a proof.” (229:14)

Timestamps of Major Segments

• [16:58] – Hamkins defines infinity & its history
• [23:04] – Hilbert’s Hotel explained
• [35:00] – Rational, real numbers, and uncountable infinity
• [48:05] – Set theory as the foundation of mathematics
• [53:04] – Axiom of choice and ZFC axioms
• [66:57] – Russell’s paradox, committees, and fruit salad analogy
• [76:09] – Gödel’s incompleteness and Hilbert’s program
• [93:44] – Truth vs. proof distinction
• [104:53] – Halting problem and diagonal arguments
• [142:38] – Continuum hypothesis and Cantor’s obsession
• [157:54] – Independence of CH, Gödel, and Cohen’s work
• [161:20] – Forcing, set-theoretic multiverse, and philosophical implications
• [180:42] – Surreal numbers and Conway’s vision
• [220:21] – Infinite chess and its mathematical challenges
• [229:14] – AI and mathematics: possibility and skepticism
• [238:32] – Favorite mathematical and philosophical ideas

Tone and Language

The conversation is enthusiastic, warm, and intellectually playful. Hamkins uses anthropomorphism and accessible analogies without sacrificing rigor. Lex maintains a tone of curiosity and wonder, alternating between technical depth and big-picture reflection.

For Further Exploration

• Joel David Hamkins: Books, blog (“Infinitely More”), Math Overflow profile
• Lex Fridman Podcast: Podcast Homepage
• Set Theory Foundations: See ZFC axioms, Cantor’s works
• Infinite Chess: Research papers by Hamkins and colleagues

End of Summary.