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Lectures on Infinity

Joel David Hamkins

Every chapter in the course, with links to the moment it begins in the video and in the transcript. For topics across all courses, see the Topic Index.

Intro Infinity Has a Story to Tell

Welcome to the Infinity Series
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Paradoxes and Conundrums of Infinity
▶︎ 0:05 transcript ↗︎
Potentially Infinite vs. Actually Infinite
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Countable vs. Uncountable Infinity
▶︎ 0:18 transcript ↗︎
From Zeno to Gödel: A Historical Journey
▶︎ 0:25 transcript ↗︎
Genuine Progress in Understanding Infinity
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Join the Lecture Series
▶︎ 1:03 transcript ↗︎

Lecture 1 Zeno's Paradox and Infinite Sums

Welcome to Infinity
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Zeno's Paradox: Why All Motion Is Impossible
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Achilles vs the Tortoise
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What Are Supertasks?
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Turning Zeno into an Infinite Sum
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Does 0.999... Really Equal 1?
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What Decimal Notation Actually Means
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Deriving the Geometric Series Formula
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When Do Infinite Sums Converge?
▶︎ 19:51 transcript ↗︎
Applying the Formula to Zeno and 0.999...
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The Harmonic Series Blows Up
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The Alternating Harmonic Series
▶︎ 31:26 transcript ↗︎
Rearranging Terms Changes the Answer
▶︎ 35:28 transcript ↗︎
Recap and What's Next: Supertasks
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Lecture 2 Supertasks: Doing Infinitely Many Things

What Are Supertasks?
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Thomson's Lamp: On or Off After Infinity?
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Zeno's Paradox Makes Supertasks Real
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Trivial Supertasks Hide in Plain Sight
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The Deal with the Devil
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Balls in a Sack: Empty After Infinity?
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Controlling What Remains in the Sack
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Random Removal and Vanishing Probability
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Probability Zero Does Not Mean Impossible
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Every Ball vs. All Balls: A Subtle Gap
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Dartboards and Countable Additivity
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The Chocolatier's Game: Finite Servings
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Infinite Servings and the Zigzag Strategy
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Can the Glutton Win Without Memory?
▶︎ 39:32 transcript ↗︎
Countable vs. Uncountable Creativity
▶︎ 41:56 transcript ↗︎
Wrapping Up Supertasks
▶︎ 45:35 transcript ↗︎

Lecture 3 The Paradox of Giants: Strange Consequences in High Dimension

Giants of Legend and Literature
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Why Physics Makes Giants Impossible
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The Square-Cube Law Destroys Giants
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Why Elephants Are Stocky and Bugs Are Thin
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The Paradox of Miniature Humans
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Evolution and Body Size Genes
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Gabriel's Horn Has Finite Volume
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Gabriel's Horn Has Infinite Surface Area
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Can You Paint an Infinite Surface?
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Extended Real Numbers and Infinity
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Testing the Paint-Based Theory of Area
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A Modified Horn with Finite Area
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Koch Snowflake Breaks the Paint Rule
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Beautiful Spirals in the Plane
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Hypersphere Volume Peaks at Dimension 5
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Why Hypercubes Are All Corners
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The Blue Sphere That Escapes Its Box
▶︎ 53:18 transcript ↗︎

Lecture 4 The Largest Tweetable Number

Can You Tweet the Biggest Possible Number?
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Why a Largest Tweetable Number Must Exist
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The Million Dollar Largest Number Contest
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Number vs. Numeral: A Giant Number One
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The Use-Mention Distinction in Language
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Zero: The Largest Number Alphabetically?
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Iterated Exponentials and Their Ambiguity
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Googolplex and Incomprehensible Numbers
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Googol Bang, Plex, and Stack Hierarchy
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Knuth's Up-Arrow Notation Explained
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Beyond Knuth: Strong Up-Arrow Notation
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What Is Kolmogorov Complexity?
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Why Kolmogorov Complexity Is Uncomputable
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The Winning Entry That Breaks Everything
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Berry's Paradox and Definability
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The Halting Problem Blocks the Judge
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Logical Independence and Axiomatic Limits
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Tarski's Theorem Resolves the Paradox
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Lecture 5 Potential vs Actual Infinity

Potential vs Actual Infinity
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Dividing a Line Segment Infinitely
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How Infinite Is a Line in Geometry?
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Archimedes and Infinite Sums
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Infinitely Many Primes: Two Proofs
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Galileo's Case for Actual Infinity
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What Is Ultrafinitism?
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The Famous Draw-the-Line Challenge
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Formalizing Ultrafinitism in Arithmetic
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Epistemological vs Ontological Potentialism
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Modal Logic Meets Potentialism
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Aristotelian Potentialism in Modal Terms
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Nonlinear Potentialism and Arbitrary Sets
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S4 Modal Axioms All Potentialists Share
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Distinguishing Potentialisms with S4.2 and S4.3
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Linear, Convergent, and Radical Branching
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The Potentialist Translation Theorem
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The Rise of Actualism in Modern Math
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Lecture 6 Galileo's Paradox of Infinity

The Puzzle of Infinite Set Sizes
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A Dinner Party and One-to-One Correspondence
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Defining Equinumerosity
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The Cantor-Hume Principle Explained
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The Paradox of Aristotle's Wheel
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Hexagonal Wheels and Vanishing Gaps
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Bolzano on Arcs, Circles, and Bijections
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Matching Points on Unequal Line Segments
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A Finite Segment Equals the Infinite Line
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Natural Numbers vs. Perfect Squares
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Reading the Original 1638 Dialogue
▶︎ 20:36 transcript ↗︎
How Modern Math Resolves the Paradox
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Comparing Sizes of Infinite Sets
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The Cantor-Schröder-Bernstein Theorem
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Lecture 7 What Does Finite Really Mean?

Is Defining 'Finite' Actually Easy?
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Aristotle's Potentialist View of Infinity
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Galileo's Paradox and Dedekind Infinite Sets
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Can Linear Orders Define Finiteness?
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Order Infinite and Linearly Finite Definitions
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Discretely Finite Sets and Counterexamples
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Stöckel Finite, Tarski Finite, and Choice
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Numerically Finite and the Circularity Problem
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Frege's Logicist Definition of Finite Number
▶︎ 17:54 transcript ↗︎
Dedekind Arithmetic and Its Three Axioms
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Why There's Only One Number System
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Finite Numbers from Dedekind's Theory
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Common, Strong, and Least Number Induction
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Every Fraction Has Lowest Terms
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Fundamental Theorem of Arithmetic by Induction
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The Pigeonhole Principle and Its Proof
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Are Dedekind Finite and Numerically Finite Equal?
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Why Numerically Finite Won
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Second-Order Logic and Expressing Finiteness
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Lecture 8 Hilbert's Hotel Is Always Open

The Infinite Hotel Thought Experiment
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Fitting One More Guest in a Full Hotel
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A Thousand New Guests Arrive
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The Infinite Bus Problem
▶︎ 3:10 transcript ↗︎
The Infinite Train with Infinite Cars
▶︎ 6:51 transcript ↗︎
Half Marathon Runners with Fraction Bibs
▶︎ 11:35 transcript ↗︎
What Does Countable Actually Mean?
▶︎ 15:22 transcript ↗︎
Closure Properties of Countable Sets
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Why the Integers Are Countable
▶︎ 20:56 transcript ↗︎
Counting Pairs with a Diagonal Path
▶︎ 22:40 transcript ↗︎
The Cantor Pairing Function
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Rational Numbers Are Countable
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Countable Union of Countable Sets
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The Hidden Role of the Axiom of Choice
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Cartesian Products Stay Countable
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All Finite Words over a Countable Alphabet
▶︎ 39:24 transcript ↗︎
Cantor's Cruise Ship and the Real Numbers
▶︎ 49:34 transcript ↗︎

Lecture 9 Beyond Countable Infinity

Review of Countable Infinity and Hilbert's Hotel
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Discovering Uncountable Infinity
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The Diagonal Argument for Uncountability
▶︎ 2:55 transcript ↗︎
Subtleties of Decimal Representations
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Worked Example of the Diagonal Construction
▶︎ 11:51 transcript ↗︎
Why It's Called the Diagonal Argument
▶︎ 16:10 transcript ↗︎
The Original 1874 Nested Interval Proof
▶︎ 17:48 transcript ↗︎
Digits and Intervals Are the Same Idea
▶︎ 23:14 transcript ↗︎
Crank Objections to the Proof
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Potential vs. Actual Infinity
▶︎ 32:04 transcript ↗︎
How the Number Concept Grew Over Time
▶︎ 34:41 transcript ↗︎
Transcendental Numbers and the Liouville Constant
▶︎ 38:40 transcript ↗︎
Proving Transcendental Numbers Must Exist
▶︎ 42:10 transcript ↗︎
Is the Proof Actually Constructive?
▶︎ 47:55 transcript ↗︎
Cantor Space and Binary Sequences
▶︎ 52:16 transcript ↗︎
Power Sets and the Size of the Reals
▶︎ 57:52 transcript ↗︎
Power Sets Are Always Strictly Larger
▶︎ 1:03:36 transcript ↗︎
An Endless Hierarchy of Infinities
▶︎ 1:10:54 transcript ↗︎
More Committees Than People, More Salads Than Fruits
▶︎ 1:15:41 transcript ↗︎
Russell's Paradox Destroys Naive Set Theory
▶︎ 1:23:05 transcript ↗︎
The Barber and Barista Paradoxes
▶︎ 1:30:35 transcript ↗︎

Lecture 10 Set Theory's Deepest Mystery

Is There an Infinity Between Integers and Reals?
▶︎ 0:00 transcript ↗︎
Ubiquitous Equinumerosity and Examples
▶︎ 3:15 transcript ↗︎
The Line Has as Many Points as the Plane
▶︎ 4:52 transcript ↗︎
Which Function Space Is Bigger?
▶︎ 11:46 transcript ↗︎
Counting Continuous Functions on the Reals
▶︎ 15:37 transcript ↗︎
The Cantor-Bendixson Process for Closed Sets
▶︎ 20:14 transcript ↗︎
Borel, Analytic, and Projective Sets
▶︎ 28:02 transcript ↗︎
Gödel's Constructible Universe and Consistency
▶︎ 30:49 transcript ↗︎
Cohen's Forcing and Independence of CH
▶︎ 34:37 transcript ↗︎
Why Large Cardinals Cannot Settle CH
▶︎ 36:49 transcript ↗︎
Universe View vs Multiverse View
▶︎ 38:57 transcript ↗︎
The Dream Solution and Why It Fails
▶︎ 42:40 transcript ↗︎
Pluralism in Geometry and Set Theory
▶︎ 46:35 transcript ↗︎