2Fast2Finite: Breaking the natural speed limit of finite numbers

To try everything Brilliant has to offer—free—for a full 30 days, visit https://brilliant.org/GSheaf/ . You’ll also get 20% off an annual premium subscription. In a previous video, we used infinite ordinals to prove that certain finite number sequences called Goodstein sequences were necessarily finite. Now, let's take this one step further and derive a formula for computing the precise length of these sequences! This will also give a bit of insight to why the previous video delved into a language of "infinities" even though the problem is purely of finite nature. Prerequisite:    • Solving a finite number problem using infi...   References: [Cai07] A.E. Caicedo. 2007. "Goodstein's function." [Cic83] E.A. Cichon. 1983. "A short proof of two recently discovered independence results using recursion theoretic methods." Proc. of the Am. Math. Soc. 87(4), 704--706. [Wai70] S.S. Wainer. 1970. "A Classification of the Ordinal Recursive Functions." Arch. Math. Logik 13, 136--153. __________ Timestamps: 00:00 - Introduction 01:15 - Recap 02:36 - A closer look at each term's shape 03:26 - omega minus one 04:04 - Wainer fundamental sequences 06:43 - Ordinal "predecessor" 08:48 - The main focus 09:41 - Fast-growing hierarchy 11:25 - The main theorem 12:19 - Induction base case 12:38 - Induction step 15:03 - Formula for Goodstein sequence length 15:50 - Thx 4 watching 16:00 - Epilogue __________ This video was sponsored by Brilliant.