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 - [ Ссылка ]
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Music used in this video:
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Daystar - Shangri-La: [ Ссылка ]
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Way Home
"Tokyo Music Walker - Way Home" is under a Creative Commons (CC-BY) license.
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Music promoted by BreakingCopyright: [ Ссылка ]
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butter by LukremBo: [ Ссылка ]
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Bensound - Enigmatic: [ Ссылка ]
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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: [ Ссылка ]
Goodstein Calculator, GitHub: [ Ссылка ]
Goodstein's Theorem, Wikipedia: [ Ссылка ]
Goodstein's Theorem, and Unprovability: [ Ссылка ]
Graham's Number - Numberphile, Youtube: [ Ссылка ]
Ordinal Number, Wikipedia: [ Ссылка ]
Ordinal Number, Wolfram MathWorld: [ Ссылка ]
Set (mathematics), Wikipedia: [ Ссылка ]
The Mindblowing Goodstein Sequences: [ Ссылка ]
Totally Ordered Set, Wolfram MathWorld: [ Ссылка ]
Well-order, Wikipedia: [ Ссылка ]
Well Ordered Set, Wolfram MathWorld: [ Ссылка ]
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