Zorn's Lemma, The Well-Ordering Theorem, and Undefinability (Version 2.0)

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. _____________________ Last PhD Update:    • Math Phd Update: First year of math grad s...   Last Academic Talk:    • Complexity of 2 Dimensional Symbolic Space...   Last CHALK video:    • Not every voting system generates impossib...   The CHALKboard:    / chalkboard   _____________________ WHAT GEAR I USED FOR THIS VIDEO: Camera - https://amzn.to/30bJRho Lens - https://amzn.to/3fgDaPa SD card - https://amzn.to/2PbTiag Microphone - https://amzn.to/2D3KYXE Lighting Kit - https://amzn.to/3ffibMH Editing Hard Drive - https://amzn.to/2P7hSsM DISCLAIMER: Links included in this description may be affiliate links. If you purchase a product or service with the links that I include here, I may receive a small commission. There is no additional charge to you! Thank you for supporting my channel and for helping me create these free videos! Subscriber Count as of release: 9,705 (+52 🙂) #CHALK #ZornsLemma #SetTheory