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Diagonal Arguments are a powerful tool in maths, and appear in several different fundamental results, like Cantor's original Diagonal argument proof (there exist uncountable sets, or "some infinities are bigger than other infinities"), Turing's Halting Problem, Gödel's incompleteness theorems, Russell's Paradox, the Liar Paradox, and even the Y Combinator. In this video, I try and motivate what a general diagonal argument looks like, from first principles. It should be accessible to anyone who's comfortable with functions and sets. The main result that I'm secretly building up towards is Lawvere's theorem in Category Theory [https://link.springer.com/chapter/10....] with inspiration from this motivating paper by Yanofsky [https://www.jstor.org/stable/3109884]. This video will be followed by a more detailed video on just Gödel's incompleteness theorems, building on the idea from this video. ====Timestamps==== 00:00 Introduction 00:59 A first look at uncountability 05:04 Why generalise? 06:53 Mathematical patterns 07:40 Working with functions and sets 11:40 Second version of Cantor's Proof 13:40 Powersets and Cantor's theorem in its generality 15:38 Proof template of Diagonal Argument 16:40 The world of Computers 21:05 Gödel numbering 23:05 An amazing program (setup of the Halting Problem) 25:05 Solution to the Halting Problem 29:49 Comparing two diagonal arguments 31:13 Lawvere's theorem 32:49 Diagonal function as a way for encoding self-reference 35:11 Summary of video 35:44 Bonus treat - Russell's Paradox CORRECTIONS 21:49 - I pronounce "Gödel" incorrectly throughout the video, sorry! Thanks to those who have pointed it out. Let me know if you spot anything else! This video has been submitted to the 3Blue1Brown Summer of Maths Exposition 2 #some2 #mathematics #maths