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

Описание: 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/

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Daystar - Shangri-La:    • ✨샛별 - 무릉도원(piano ver.)✨(Royalty Free Music)  
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Way Home
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butter by LukremBo:    • lukrembo - butter (no copyright music)  
onion by LukremBo:    • lukrembo - onion (no copyright music)  
wine by LukremBo:    • lukrembo - wine (no copyright music)  
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Bensound - Enigmatic:    • Bensound: "Enigmatic" - Royalty Free Music  
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References
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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...
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