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:    • AIRGLOW – New Touch [Synthwave] 🎵 from Roy...   Synthwave E:    • Synthwave E - Royalty Free Music   Daystar - Shangri-La:    • ✨샛별 - 무릉도원(piano ver.)✨(Royalty Free Music)   ------------------------------------------ Way Home "Tokyo Music Walker - Way Home" is under a Creative Commons (CC-BY) license.    / @tokyomusicwalker4038   Music promoted by BreakingCopyright: https://bit.ly/way-home-song ------------------------------------------ butter by LukremBo:    • lukrembo - butter   onion by LukremBo:    • lukrembo - onion   wine by LukremBo:    • lukrembo - wine   Music from https://freetousemusic.com ------------------------------------------ Bensound - Enigmatic:    • Bensound: "Enigmatic" - Royalty Free Music   ------------------------------------------ ------------------------------------------ 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/Ackerma... Goodstein Calculator, GitHub: https://github.com/WGUNDERWOOD/goodst... Goodstein's Theorem, Wikipedia: https://en.wikipedia.org/wiki/Goodste... Goodstein's Theorem, and Unprovability: https://www.sas.upenn.edu/~htowsner/G... Graham's Number - Numberphile, Youtube:    • Graham's Number - Numberphile   Ordinal Number, Wikipedia: https://en.wikipedia.org/wiki/Ordinal... Ordinal Number, Wolfram MathWorld: http://mathworld.wolfram.com/OrdinalN... Set (mathematics), Wikipedia: https://en.wikipedia.org/wiki/Set_(ma...) The Mindblowing Goodstein Sequences: https://risingentropy.com/the-mindblo... Totally Ordered Set, Wolfram MathWorld: http://mathworld.wolfram.com/TotallyO... Well-order, Wikipedia: https://en.wikipedia.org/wiki/Well-order Well Ordered Set, Wolfram MathWorld: http://mathworld.wolfram.com/WellOrde... ------------------------------------------