The sequence that grows remarkably large, then drops to zero!

Goodstein sequences can get larger than Graham's number and the growth rate can be faster than Ackermann’s function. In fact, these sequences grow at such an incredible rate, that the theorem literally cannot be proven using first order arithmetic and can only be proven using a stronger system – namely second order arithmetic. Despite this, all Goodstein sequences eventually terminate (Goodstein’s Theorem). This video will attempt to define and prove Goodstein's Theorem. This is my submission for the 3Blue1Brown Summer of Math Exposition 2 event. #Goodstein #SoME2 Some of the math animations used in this video was created using Manim - https://www.manim.community/ ------------------------------------------ Music used in this video: ------------------------------------------ AIRGLOW – New Touch: https://youtu.be/r_Q15eu03z8 Synthwave E: https://www.youtube.com/watch?v=RBxnYXGNNAk Daystar - Shangri-La: https://youtu.be/-J-ey1JZjUE ------------------------------------------ Way Home "Tokyo Music Walker - Way Home" is under a Creative Commons (CC-BY) license. https://www.youtube.com/channel/UC3lLfvhpPGtwd5qD25cMDcA Music promoted by BreakingCopyright: https://bit.ly/way-home-song ------------------------------------------ butter by LukremBo: https://www.youtube.com/watch?v=Ua7Qfc1xu90 onion by LukremBo: https://www.youtube.com/watch?v=KGQNrzqrGqw wine by LukremBo: https://www.youtube.com/watch?v=xjSxWMbz_FQ Music from https://freetousemusic.com ------------------------------------------ Bensound - Enigmatic: https://www.youtube.com/watch?v=DHD0TesReqo ------------------------------------------ ------------------------------------------ References ------------------------------------------ C. Taylor, True But Not Provable. AMSI, Melbourne, 2013. P. R. Halmos, Naive Set Theory. Springer, 1974. R. Michael, Goodstein's theorem revisted. Leeds, 2014 Ackermann function, Wikipedia: https://en.wikipedia.org/wiki/Ackermann_function Goodstein Calculator, GitHub: https://github.com/WGUNDERWOOD/goodstein-calculator Goodstein's Theorem, Wikipedia: https://en.wikipedia.org/wiki/Goodstein%27s_theorem Goodstein's Theorem, and Unprovability: https://www.sas.upenn.edu/~htowsner/GoodsteinsTheorem.pdf Graham's Number - Numberphile, Youtube: https://www.youtube.com/watch?v=XTeJ64KD5cg Ordinal Number, Wikipedia: https://en.wikipedia.org/wiki/Ordinal_number Ordinal Number, Wolfram MathWorld: http://mathworld.wolfram.com/OrdinalNumber.html Set (mathematics), Wikipedia: https://en.wikipedia.org/wiki/Set_(mathematics) The Mindblowing Goodstein Sequences: https://risingentropy.com/the-mindblowing-goodstein-sequences/ Totally Ordered Set, Wolfram MathWorld: http://mathworld.wolfram.com/TotallyOrderedSet.html Well-order, Wikipedia: https://en.wikipedia.org/wiki/Well-order Well Ordered Set, Wolfram MathWorld: http://mathworld.wolfram.com/WellOrderedSet.html ------------------------------------------