Understanding Zorn's Lemma: A Clear Explanation

Zorn's lemma states that in a partially ordered set, if every chain has an upper bound, then the set contains a maximal element. But what in the world does that mean? Many people have been confused by Zorn's lemma, and the fact that it's mainly mentioned just in the context of the axiom of choice only makes the problem worse. In this video, I dispel the confusion around Zorn's lemma by showing what it's actually saying and when it's used. This video is my #SoMEπ submission, and just in time too.

Links:
The simple proof of Zorn's lemma: https://digitalcommons.kennesaw.edu/cgi/viewcontent.cgi?article=2161&context=facpubs
My own write-up of it, since I ended up changing more than I initially thought: https://drive.google.com/file/d/1aRHrLP1WtYkmLqhPMokDmWVdECHsUoHD/view
My formalization of it: https://github.com/sudgy/math-from-nothing/blob/master/src/Set/zorn.v
My construction of the real numbers: https://drive.google.com/file/d/1fb1pNJVMuROmpLq-kDwXXeY2QxzCV0hy/view
My formalization of it: https://github.com/sudgy/math-from-nothing/tree/master/src/Number/Real/Zorn

Discord: https://discord.gg/3Zj59zA2Rg

Patreon: https://www.patreon.com/sudgylacmoe

Supporters:
David Johnston
Jason Killian
jerrud
p11
Richard Penner
trb

Sections:
00:00 Introduction
01:24 Basis Introduction
01:48 Basis Argument
03:14 Well Order Introduction
06:23 Well Order Argument
07:47 Real Number Introduction
09:27 Real Number Argument
10:48 Stating Zorn's Lemma
14:13 Using Zorn's Lemma
16:37 Basis Proof
18:26 Well Order Proof
21:30 Real Number Proof
25:30 Proof of Zorn's Lemma
32:44 Conclusion