Zorn's Lemma and The Well-ordering Theorem are seemingly straightforward statements, but they give incredibly mind-bending results. Orderings, Hasse Diagrams, and the Ordinals / set theory will come up in this video as tools to get a better view of where the "proof" of Zorn's lemma comes from. Accepting the Axiom of Choice, which turns out to be equivalent to both Zorn's lemma and the Well-Ordering Theorem actually introduces a notion of an undefinable well-ordering; an order paradox -- specifically an impossibility of ordering the real numbers. The video can be broken up into the following sections:
00:00 Intro
00:48 Relations 101: Equality
02:52 Partial Order Definition and Examples
04:04 Hasse Diagram
05:12 Powersets and The Subset Partial Order
08:07 Zorn's Lemma
08:56 Ordinals in Brief
09:50 Sketch of Zorn's Lemma
12:18 Zorn's Lemma, The Well-Ordering Theorem, and The Axiom of Choice
13:06 The Well-Ordering Theorem and Undefinability
***Note in the section on Chains, two different chains are given even though I sound like I am talking about only one.
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