What are the natural numbers? The integers? The rationals? The reals? While we may have an intuitive understanding of these numbers and sets, it is not so easy to actually construct these sets formally. To do so, we must use some axioms of set theory, and using only these assumptions, formally describe what these infinite sets should look like. We will develop various tools in set theory, like ordered pairs, relations, ordering, and equivalence classes, to begin with only zero, and from nothing, build all of the real numbers.
0:00 Introduction
1:46 Set Theory and Basic Notions
8:13 Axiom of Infinity and the Naturals
13:09 The Integers
23:19 The Rationals
26:00 The Reals
36:38 Conclusion
Additional Resources:
Wikipedia article on the Construction of the naturals: https://en.wikipedia.org/w/index.php?title=Natural_number#Set-theoretic_definition
Wikipedia article on the Construction of the Reals: https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
Wikipedia article on ZFC: https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory
Axiom of Choice video: https://youtu.be/szfsGJ_PGQ0
Cardinality of the Continuum video: https://youtu.be/iaUwNuaSLUk
Music:
https://c418.bandcamp.com/album/dief
Imaginary Interlude by C418
https://c418.bandcamp.com/album/circle
minimal by C418
love by C418
https://patriciataxxon.bandcamp.com/album/crocus
Crocus 2 by Patricia Taxxon
Far the Days Come by Letter Box
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Animations were made by Manim, an open-source python-based animation program by 3Blue1Brown.
https://github.com/3b1b/manim
This video was submitted to 3Blue1Brown's SoMEπ (Summer of Math Exposition Community Edition).
https://some.3b1b.co/