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Goodstein's theorem
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== Sequence length as a function of the starting value == The '''Goodstein function''', <math>\mathcal{G}: \mathbb{N} \to \mathbb{N} </math>, is defined such that <math>\mathcal{G}(n)</math> is the length of the Goodstein sequence that starts with ''n''. (This is a [[total function]] since every Goodstein sequence terminates.) The extremely high growth rate of <math>\mathcal{G}</math> can be calibrated by relating it to various standard ordinal-indexed hierarchies of functions, such as the functions <math>H_\alpha</math> in the [[Hardy hierarchy]], and the functions <math>f_\alpha</math> in the [[fast-growing hierarchy]] of LΓΆb and Wainer: * Kirby and Paris (1982) proved that :<math>\mathcal{G}</math> has approximately the same growth-rate as <math>H_{\epsilon_0}</math> (which is the same as that of <math>f_{\epsilon_0}</math>); more precisely, <math>\mathcal{G}</math> dominates <math>H_\alpha</math> for every <math>\alpha < \epsilon_0</math>, and <math>H_{\epsilon_0}</math> dominates <math>\mathcal{G}\,\!.</math> :(For any two functions <math>f, g: \mathbb{N} \to \mathbb{N} </math>, <math>f</math> is said to dominate <math>g</math> if <math>f(n) > g(n)</math> for all sufficiently large <math>n</math>.) * Cichon (1983) showed that :<math> \mathcal{G}(n) = H_{R_2^\omega(n+1)}(1) - 1, </math> :where <math>R_2^\omega(n)</math> is the result of putting ''n'' in hereditary base-2 notation and then replacing all 2s with Ο (as was done in the proof of Goodstein's theorem). * Caicedo (2007) showed that if <math> n = 2^{m_1} + 2^{m_2} + \cdots + 2^{m_k} </math> with <math> m_1 > m_2 > \cdots > m_k, </math> then :<math> \mathcal{G}(n) = f_{R_2^\omega(m_1)}(f_{R_2^\omega(m_2)}(\cdots(f_{R_2^\omega(m_k)}(3))\cdots)) - 2</math>. Some examples: {| class="wikitable" border="1" |- ! colspan=3 | n ! colspan=3 | <math>\mathcal{G}(n)</math> |- | 1 | <math>2^0</math> | <math>2 - 1</math> | <math>H_\omega(1) - 1</math> | <math>f_0(3) - 2</math> | 2 |- | 2 | <math>2^1</math> | <math>2^1 + 1 - 1</math> | <math>H_{\omega + 1}(1) - 1</math> | <math>f_1(3) - 2</math> | 4 |- | 3 | <math>2^1 + 2^0</math> | <math>2^2 - 1</math> | <math>H_{\omega^\omega}(1) - 1</math> | <math>f_1(f_0(3)) - 2</math> | 6 |- | 4 | <math>2^2</math> | <math>2^2 + 1 - 1</math> | <math>H_{\omega^\omega + 1}(1) - 1</math> | <math>f_\omega(3) - 2</math> | 3Β·2<sup>402653211</sup> β 2 β 6.895080803Γ10<sup>121210694</sup> |- | 5 | <math>2^2 + 2^0</math> | <math>2^2 + 2 - 1</math> | <math>H_{\omega^\omega + \omega}(1) - 1</math> | <math>f_\omega(f_0(3)) - 2</math> | > [[Ackermann function|''A'']](4,4) > 10<sup>10<sup>10<sup>19727</sup></sup></sup> |- | 6 | <math>2^2 + 2^1</math> | <math>2^2 + 2 + 1 - 1</math> | <math>H_{\omega^\omega + \omega + 1}(1) - 1</math> | <math>f_\omega(f_1(3)) - 2</math> | > ''A''(6,6) |- | 7 | <math>2^2 + 2^1 + 2^0</math> | <math>2^{2 + 1} - 1</math> | <math>H_{\omega^{\omega + 1}}(1) - 1</math> | <math>f_\omega(f_1(f_0(3))) - 2</math> | > ''A''(8,8) |- | 8 | <math>2^{2 + 1}</math> | <math>2^{2 + 1} + 1 - 1</math> | <math>H_{\omega^{\omega + 1} + 1}(1) - 1</math> | <math>f_{\omega + 1}(3) - 2</math> | > ''A''<sup>3</sup>(3,3) = ''A''(''A''(61, 61), ''A''(61, 61)) |- | colspan=6 align=center | <math>\vdots</math> |- | 12 | <math>2^{2 + 1} + 2^2</math> | <math>2^{2 + 1} + 2^2 + 1 - 1</math> | <math>H_{\omega^{\omega + 1} + \omega^\omega + 1}(1) - 1</math> | <math>f_{\omega + 1}(f_\omega(3)) - 2</math> | > ''f''<sub>Ο+1</sub>(64) > [[Graham's number]] |- | colspan=6 align=center | <math>\vdots</math> |- | 19 | <math>2^{2^2} + 2^1 + 2^0</math> | <math>2^{2^2} + 2^2 - 1</math> | <math>H_{\omega^{\omega^\omega} + \omega^\omega}(1) - 1</math> | <math>f_{\omega^\omega}(f_1(f_0(3))) - 2</math> | |- |} (For [[Ackermann function]] and [[Graham's number]] bounds see [[fast-growing hierarchy#Functions in fast-growing hierarchies]].)
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