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