The Axiom that Shapes Infinity

This is one of the most fundamental statements in set theory. It allows us to imagine incredibly powerful objects with great precision. It molds the infinity known as the continuum, allowing us to play with difficult sets like the real numbers. It is a statement that impacts fields like analysis, measure theory, combinatorics, and topology. Finally, it stems from a technique of set theory, called forcing, which is one of the most powerful and interesting concepts in all of mathematical logic, allowing us to construct incredible mathematical models of a vast set of axioms. We will study the statement known as Martin's Axiom, which is useful not only in its applications to the world of mathematics but also as a great introduction to the more advanced aspects of set theory.

0:00 Introduction
2:16 The Bounding Number Problem
6:33 Posets
15:50 Martin's Axiom
24:20 MA and the Bounding Number
27:21 MA and the Lebesgue Measure
29:30 MA and Suslin's Problem
31:36 Conclusion

Additional Resources:
Wikipedia article on Martin's Axiom: https://en.wikipedia.org/wiki/Martin'...
Wikipedia article on the bounding number and similar cardinals: https://en.wikipedia.org/wiki/Cardina...
Wikipedia article on the Lebesgue measure: https://en.wikipedia.org/wiki/Lebesgu...
Wikipedia article on Suslin's Problem: https://en.wikipedia.org/wiki/Suslin%...
Set Theory: An Introduction to Independence Proofs by Kenneth Kunen: the university-standard textbook for advanced set theory
All the proofs in this video, formalized: https://drive.google.com/file/d/1k_ei...

Music, by order of appearance:
Sentimental - White Bat Audio/Karl Casey
Core - White Bat Audio/Karl Casey
Icarus - White Bat Audio/Karl Casey
Synthetic Dreams - White Bat Audio/Karl Casey
26 - C418
Life changing moments seem minor in pictures - C418
sleepless - C418
Work Life Imbalance - C418

White Bat Audio/Karl Casey: https://karlcasey.bandcamp.com/
C418: https://c418.bandcamp.com/

Animations were made using ManimCE, an community-maintained version of manim, an open-source python-based animation program by 3Blue1Brown.
https://www.manim.community/
https://github.com/3b1b/manim

This video was submitted to 3Blue1Brown's SoME4 (Summer of Math Exposition).
https://some.3b1b.co/