Cantor's proof that the real numbers are uncountable

First, it will be explained, what mathematicians mean, when they talk about countable sets, even when they have infinitely many elements.
In 1874, Cantor proofed that the set of all real numbers is even larger than a countably infinite set. Which means that one cannot count through all the real numbers. The illustration of Cantor’s proof is the main theme of this video.
The construction that Cantor used in his proof is finally applied the countable set of fractions to shed more light on the ideas behind it. As a byproduct irrational numbers are discovered as a limit of a continued fraction expansion.

0:00 - Intro
0:48 - Finite sets
1:15 - Countable sets
2:52 - The set of all fractions
4:12 - Fractions and the number line
4:45 - Cantor's proof of Uncountability
5:11 - The construction of the proof
6:48 - Summary
7:28 - An Example for the construction
8:08 - The result: Irrational numbers
9:42 - Take-away
10:28 - Continued Fraction expansion for 1/sqrt(2)

#cantor #continuedfractionexpansion #irrationalnumbers #countable #uncountable #settheory

All animations in this video have been created by the manim library.
https://github.com/3b1b/manim

The inspiration for the topic arose from the introduction of the book:
Hoffmann, Dirk: Limits of mathematics. A journey through the core subjects of mathematical logic.

The source for the animations are available under:
https://github.com/mathelehrer/manim/...