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Course

Lectures on Infinity

Chapters

Intro Infinity Has a Story to Tell

Welcome to the Infinity Series
0:01
Paradoxes and Conundrums of Infinity
0:05
Potentially Infinite vs. Actually Infinite
0:14
Countable vs. Uncountable Infinity
0:18
From Zeno to Gödel: A Historical Journey
0:25
Genuine Progress in Understanding Infinity
0:41
Join the Lecture Series
1:03

Lecture 1 Zeno's Paradox and Infinite Sums

Welcome to Infinity
0:00
Zeno's Paradox: Why All Motion Is Impossible
0:13
Achilles vs the Tortoise
2:07
What Are Supertasks?
3:57
Turning Zeno into an Infinite Sum
4:16
Does 0.999... Really Equal 1?
6:34
What Decimal Notation Actually Means
10:32
Deriving the Geometric Series Formula
14:26
When Do Infinite Sums Converge?
19:51
Applying the Formula to Zeno and 0.999...
23:13
The Harmonic Series Blows Up
25:11
The Alternating Harmonic Series
31:26
Rearranging Terms Changes the Answer
35:28
Recap and What's Next: Supertasks
40:38

Lecture 2 Supertasks: Doing Infinite Things

What Are Supertasks?
0:00
Thompson's Lamp: On or Off After Infinity?
0:10
Zeno's Paradox Makes Supertasks Real
4:00
Trivial Supertasks Hide in Plain Sight
5:03
The Deal with the Devil
6:22
Balls in a Sack: Empty After Infinity?
12:03
Controlling What Remains in the Sack
16:04
Random Removal and Vanishing Probability
17:01
Probability Zero Does Not Mean Impossible
21:18
Every Ball vs. All Balls: A Subtle Gap
23:40
Dartboards and Countable Additivity
25:00
The Chocolatier's Game: Finite Servings
30:12
Infinite Servings and the Zigzag Strategy
35:31
Can the Glutton Win Without Memory?
39:32
Countable vs. Uncountable Creativity
41:56
Wrapping Up Supertasks
45:35

Lecture 3 Volume, Surface, and the Infinite

Giants of Legend and Literature
0:02
Why Physics Makes Giants Impossible
1:47
The Square-Cube Law Destroys Giants
5:31
Why Elephants Are Stocky and Bugs Are Thin
7:22
The Paradox of Miniature Humans
8:53
Evolution and Body Size Genes
10:09
Gabriel's Horn Has Finite Volume
11:56
Gabriel's Horn Has Infinite Surface Area
16:10
Can You Paint an Infinite Surface?
18:20
Extended Real Numbers and Infinity
20:55
Testing the Paint-Based Theory of Area
25:05
A Modified Horn with Finite Area
27:34
Koch Snowflake Breaks the Paint Rule
30:32
Beautiful Spirals in the Plane
37:47
Hypersphere Volume Peaks at Dimension 5
40:50
Why Hypercubes Are All Corners
47:27
The Blue Sphere That Escapes Its Box
53:18

Lecture 4 The Largest Tweetable Number

Can You Tweet the Biggest Possible Number?
0:00
Why a Largest Tweetable Number Must Exist
1:43
The Million Dollar Largest Number Contest
5:01
Number vs. Numeral: A Giant Number One
6:31
The Use-Mention Distinction in Language
8:20
Zero: The Largest Number Alphabetically?
11:58
Iterated Exponentials and Their Ambiguity
13:16
Googolplex and Incomprehensible Numbers
15:45
Googol Bang, Plex, and Stack Hierarchy
20:00
Knuth's Up-Arrow Notation Explained
24:20
Beyond Knuth: Strong Up-Arrow Notation
28:00
What Is Kolmogorov Complexity?
29:57
Why Kolmogorov Complexity Is Uncomputable
32:11
The Winning Entry That Breaks Everything
34:42
Berry's Paradox and Definability
37:28
The Halting Problem Blocks the Judge
41:54
Logical Independence and Axiomatic Limits
44:18
Tarski's Theorem Resolves the Paradox
47:43

Lecture 5 Potential vs Actual Infinity

Potential vs Actual Infinity
0:01
Dividing a Line Segment Infinitely
2:00
How Infinite Is a Line in Geometry?
3:20
Archimedes and Infinite Sums
4:53
Infinitely Many Primes: Two Proofs
7:59
Galileo's Case for Actual Infinity
12:51
What Is Ultrafinitism?
16:05
The Famous Draw-the-Line Challenge
18:11
Formalizing Ultrafinitism in Arithmetic
20:15
Epistemological vs Ontological Potentialism
24:29
Modal Logic Meets Potentialism
25:26
Aristotelian Potentialism in Modal Terms
28:49
Nonlinear Potentialism and Arbitrary Sets
33:00
S4 Modal Axioms All Potentialists Share
37:04
Distinguishing Potentialisms with S4.2 and S4.3
41:55
Linear, Convergent, and Radical Branching
48:50
The Potentialist Translation Theorem
51:02
The Rise of Actualism in Modern Math
57:15

Lecture 6 Galileo's Paradox of Infinity

The Puzzle of Infinite Set Sizes
0:00
A Dinner Party and One-to-One Correspondence
0:24
Defining Equinumerosity
1:07
The Cantor-Hume Principle Explained
2:29
The Paradox of Aristotle's Wheel
4:02
Hexagonal Wheels and Vanishing Gaps
8:16
Bolzano on Arcs, Circles, and Bijections
12:21
Matching Points on Unequal Line Segments
14:06
A Finite Segment Equals the Infinite Line
15:35
Natural Numbers vs. Perfect Squares
17:32
Reading the Original 1638 Dialogue
20:36
How Modern Math Resolves the Paradox
23:31
Comparing Sizes of Infinite Sets
25:38
The Cantor-Schröder-Bernstein Theorem
28:48

Lecture 7 What Does Finite Really Mean?

Is Defining 'Finite' Actually Easy?
0:00
Aristotle's Potentialist View of Infinity
1:02
Galileo's Paradox and Dedekind Infinite Sets
1:38
Can Linear Orders Define Finiteness?
3:51
Order Infinite and Linearly Finite Definitions
6:18
Discretely Finite Sets and Counterexamples
10:57
Stöckel Finite, Tarski Finite, and Choice
13:17
Numerically Finite and the Circularity Problem
15:52
Frege's Logicist Definition of Finite Number
17:54
Dedekind Arithmetic and Its Three Axioms
20:56
Why There's Only One Number System
23:38
Finite Numbers from Dedekind's Theory
28:55
Common, Strong, and Least Number Induction
30:38
Every Fraction Has Lowest Terms
37:00
Fundamental Theorem of Arithmetic by Induction
38:38
The Pigeonhole Principle and Its Proof
41:02
Are Dedekind Finite and Numerically Finite Equal?
44:39
Why Numerically Finite Won
50:17
Second-Order Logic and Expressing Finiteness
52:04

Lecture 8 Hilbert's Hotel Is Always Open

The Infinite Hotel Thought Experiment
0:01
Fitting One More Guest in a Full Hotel
0:37
A Thousand New Guests Arrive
2:05
The Infinite Bus Problem
3:10
The Infinite Train with Infinite Cars
6:51
Half Marathon Runners with Fraction Bibs
11:35
What Does Countable Actually Mean?
15:22
Closure Properties of Countable Sets
17:39
Why the Integers Are Countable
20:56
Counting Pairs with a Diagonal Path
22:40
The Cantor Pairing Function
26:12
Rational Numbers Are Countable
31:10
Countable Union of Countable Sets
32:20
The Hidden Role of the Axiom of Choice
34:57
Cartesian Products Stay Countable
37:27
All Finite Words over a Countable Alphabet
39:24
Cantor's Cruise Ship and the Real Numbers
49:34

Lecture 9 Beyond Countable Infinity

Review of Countable Infinity and Hilbert's Hotel
0:00
Discovering Uncountable Infinity
1:35
The Diagonal Argument for Uncountability
2:55
Subtleties of Decimal Representations
8:01
Worked Example of the Diagonal Construction
11:51
Why It's Called the Diagonal Argument
16:10
The Original 1874 Nested Interval Proof
17:48
Digits and Intervals Are the Same Idea
23:14
Crank Objections to the Proof
28:39
Potential vs. Actual Infinity
32:04
How the Number Concept Grew Over Time
34:41
Transcendental Numbers and the Liouville Constant
38:40
Proving Transcendental Numbers Must Exist
42:10
Is the Proof Actually Constructive?
47:55
Cantor Space and Binary Sequences
52:16
Power Sets and the Size of the Reals
57:52
Power Sets Are Always Strictly Larger
1:03:36
An Endless Hierarchy of Infinities
1:10:54
More Committees Than People, More Salads Than Fruits
1:15:41
Russell's Paradox Destroys Naive Set Theory
1:23:05
The Barber and Barista Paradoxes
1:30:35

Lecture 10 Set Theory's Deepest Mystery

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