FANDOM


R_{a}^{b}(n) : replace a to b in base-a hereditary representation of n

R_{a}^{b}(n)=R_{a}^{b}(n,0)
R_{a}^{b}(0,t)=0
R_{a}^{b}(n,t)=R_{a}^{b}(\lfloor n/a \rfloor,t+1)+b^{R_{a}^{b}(t,0)}(n-a\lfloor n/a \rfloor))

g_{\alpha,n} : Goodstein sequence starting with n and bumping base b to G_{\alpha}(b)

g_{\alpha,n}(0)=n
g_{\alpha,n}(m+1)=max(R_{{G_{\alpha}}^{m}(2)}^{{G_{\alpha}}^{m+1}(2)}(g_{\alpha,n}(m)),1)-1

G_{0}(n)=n+1

G_{\alpha+1}(n) : the base when g_{\alpha,n} terminates

G_{\alpha+1}(n)={G_{\alpha}}^{\mu k(g_{\alpha,n}(k)=0)}(2)

G_{\alpha}(n)=G_{\alpha \left [ n \right ]}(n) if \alpha is a limit ordinal


example:

R_{2}^{5}(11)=R_{2}^{5}(2^{2+1}+2+1)=5^{5+1}+5+1=15631
G_{1}(3)=7
n base g_{0,3}(n)
0 2 2+1
1 3 3
2 4 3
3 5 2
4 6 1
5 7 0
G_{2}(2)={G_{1}}^{5}(2)
n base g_{1,2}(n)
0 2 2
1 G_{1}(2)=5 4
2 {G_{1}}^{2}(2) 3
3 {G_{1}}^{3}(2) 2
4 {G_{1}}^{4}(2) 1
5 {G_{1}}^{5}(2) 0

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