User blog:B1mb0w/The R Function

The R Function
The R function generates very large numbers. It is based on my earlier work on the The S Function. It has a growth rate \(\approx f_{LVO}(n)\).

What is the R Function
The R Function is actually two functions \(R\) and \(r\) which this simple ruleset:

\(R(n) = R(0,n) = n + 1\)

\(R(a + 1, n) = R^n(a,n_*)\)

\(R(r(0), n) = R(n,n)\)

\(r(a + 1, n) = R^{r(a)}(r(a)_*,r(a))\)

Growth Rate of the R Function ... to \(\Gamma_0\)
The R Function behaves like the FGH function up to a point:

\(R^h(g,n_*) = f_g^h(n)\)

\(R(r(0),n) = f_{\omega}(n)\)

\(R(R(r(0),r(0)),n) \approx f_{\varphi(\omega,0)}(n)\)

\(R(R(R^2(r(0)),r(0)),n) = f_{\varphi(\omega,0)}(n)\)

\(S(n,S^2(g(0),g(0)_*,1),1) > f_{\varphi^2(1_*,0)}(n)\)

\(S(n,g(1),1) > S(n,S^{g(0)}(g(0),g(0)_*,1),1) > f_{\varphi^{\omega}(1_*,0)}(n) = f_{\varphi(1,0,0)}(n) = f_{\Gamma_0}(n)\)

Growth Rate ... to svo
The Generalised S Function will eventually reach and surpass the small Veblen ordinal (svo):

\(S(n,S(g(1),g(0),1),1) \approx f_{\varphi(\omega,\varphi(1,0,0)+1)}(n)\)

\(S(n,S(g(1),S(g(0),g(0),1),1),1) \approx f_{\varphi(\varphi(\omega,0),\varphi(1,0,0)+1)}(n)\)

\(S(n,S(g(1),g(1),1),1) \approx f_{\varphi(1,0,1)}(n)\)

\(S(n,S(g(1),S(g(1),1,1),1),1) \approx f_{\varphi(1,0,2)}(n)\)

\(S(n,S(g(1),S(g(1),2,1),1),1) \approx f_{\varphi(1,0,\omega^2)}(n)\)

\(S(n,S(g(1),S(g(1),3,1),1),1) \approx f_{\varphi(1,0,\varphi(1,0))}(n)\)

\(S(n,S^2(g(1),g(1)_*,1),1) = S(n,S(g(1),S(g(1),g(1),1),1),1) \approx f_{\varphi(1,0,\varphi(1,0,0))}(n) = f_{\varphi^2(1,0,0_*)}(n)\)

\(S(n,S^{g(0)}(g(1),g(1)_*,1),1) \approx f_{\varphi^{\omega}(1,0,0_*)}(n) = f_{\varphi(1,1,0)}(n)\)

\(S(n,S^{S(g(0),1,1)}(g(1),g(1)_*,1),1) \approx f_{\varphi(1,2,0)}(n)\)

\(S(n,g(2),1) > S(n,S^{g(1)}(g(1),g(1)_*,1),1) \approx f_{\varphi^2(1,0_*,0)}(n)\)

\(S(n,S(g(2),g(2),1),1) \approx f_{\varphi(1,\varphi(1,0,0),\varphi^2(1,0_*,0))}(n) = f_{\varphi^2(1,\varphi(1,0,0),0_*)}(n)\)

\(S(n,S^{g(0)}(g(2),g(2)_*,1),1) \approx f_{\varphi(1,\varphi(1,0,0)+1,0)}(n)\)

\(S(n,S^{g(1)}(g(2),g(2)_*,1),1) \approx f_{\varphi(1,\varphi(1,0,0).2,0)}(n)\)

\(S(n,S^{g(1)}(g(2),g(2)_*,1),1) \approx f_{\varphi(1,\varphi(1,0,0).2,0)}(n)\)

\(S(n,g(3),1) > S(n,S^{g(2)}(g(2),g(2)_*,1),1) \approx f_{\varphi^3(1,0_*,0)}(n)\)

\(S(n,g^2(0),1) = S(n,g(g(0)),1) \approx f_{\varphi(2,0,0)}(n)\)

\(S(n,g^3(0),1) \approx f_{\varphi^2(2,0,0_*)}(n)\)

\(S(n,g(1,0),1) = S(n,g^{(g(0))}(g(0)),1) \approx f_{\varphi(2,1,0)}(n)\)

\(S(n,g^{g(0)}(g(1,0)),1) \approx f_{\varphi(2,1,1)}(n)\)

\(S(n,g^{S(g(0),1,1)}(g(1,0)),1) \approx f_{\varphi(2,1,2)}(n)\)

\(S(n,g^{S(g(0),2,1)}(g(1,0)),1) \approx f_{\varphi(2,1,\omega^2)}(n)\)

\(S(n,g^{S(g(0),g(0),1)}(g(1,0)),1) \approx f_{\varphi(2,1,\varphi(\omega,0))}(n)\)

\(S(n,g^{g(1)}(g(1,0)),1) \approx f_{\varphi(2,1,\varphi(1,0,0))}(n)\)

\(S(n,g(1,1),1) = S(n,g^{g(1,0)}(g(1,0)),1) \approx f_{\varphi(2,1,\varphi(2,1,0))}(n) = f_{\varphi^2(2,1,0_*)}(n)\)

\(S(n,g(1,g(0)),1) \approx f_{\varphi(2,2,0)}(n)\)

\(S(n,g(1,g(1,0)),1) \approx f_{\varphi^2(2,0_*,0)}(n)\)

\(S(n,g^{g(0)}(1,g(1,0)_*),1) \approx f_{\varphi(3,0,0)}(n)\)

\(S(n,g^{S(g(0),0,1)}(1,g(1,0)_*),1) \approx f_{\varphi(2,\varphi(3,0,0)+1,0)}(n)\)

\(S(n,g^{S(g(0),1,1)}(1,g(1,0)_*),1) \approx f_{\varphi(3,0,1)}(n)\)

\(S(n,g^{S(g(0),2,1)}(1,g(1,0)_*),1) \approx f_{\varphi(3,0,\omega^2)}(n)\)

\(S(n,g^{g(1)}(1,g(1,0)_*),1) \approx f_{\varphi(3,0,\varphi(1,0,0))}(n)\)

\(S(n,g^{g(2)}(1,g(1,0)_*),1) \approx f_{\varphi(3,0,\varphi^2(1,0_*,0))}(n)\)

\(S(n,g(2,0),1) > S(n,g^{g(1,0)}(1,g(1,0)_*),1) \approx f_{\varphi(3,0,\varphi(2,1,0))}(n)\)

\(S(n,g(2,1),1) \approx f_{\varphi^2(3,0,0_*)}(n)\)

\(S(n,g(2,g(0)),1) \approx f_{\varphi(3,1,0)}(n)\)

\(S(n,g(2,S(g(0),1,1)),1) \approx f_{\varphi(3,1,1)}(n)\)

\(S(n,g(2,g(1)),1) \approx f_{\varphi(3,1,\varphi(1,0,0))}(n)\)

\(S(n,g^2(2,0_*),1) \approx f_{\varphi^2(3,1,0_*)}(n)\)

\(S(n,g^{g(0)}(2,0_*),1) \approx f_{\varphi(3,2,0)}(n)\)

\(S(n,g^{g(1)}(2,0_*),1) \approx f_{\varphi(3,\varphi(1,0,0),0)}(n)\)

\(S(n,g(3,0),1) > S(n,g^{g(2,0)}(2,0_*),1) \approx f_{\varphi^2(3,0_*,0)}(n)\)

\(S(n,g(3,1),1) \approx f_{\varphi^3(3,0_*,0)}(n)\)

\(S(n,g(3,g(0)),1) \approx f_{\varphi^{\omega}(3,0_*,0)}(n) = f_{\varphi(4,0,0)}(n)\)

\(S(n,g(4,1),1) \approx f_{\varphi^2(4,0_*,0)}(n)\)

\(S(n,g(4,g(0)),1) \approx f_{\varphi(5,0,0)}(n)\)

\(S(n,g(g(0),g(0)),1) \approx f_{\varphi(\omega,0,0)}(n)\)

\(S(n,g^2(g(0),g(0)),1) \approx f_{\varphi^2(1_*,0,0)}(n)\)

\(S(n,g(1,0,0),1) > S(n,g^{g(0)}(g(0),g(0)),1) \approx f_{\varphi(1,0,0,0)}(n)\)

\(S(n,g(1,0_{[g(0)]}),1) > S(n,g(1,0_{[n-1]}),1) \approx f_{\varphi(1,0_{[n]})}(n) = f_{svo}(n)\)

Growth Rate ... to LVO
The Generalised S Function is one of the Fastest Computable functions:

\(g(0) \approx \omega = \vartheta(0)\)

\(S(g(0),3,1) \approx \epsilon_0 = \varphi(1,0) = \vartheta(1)\)

\(g(1) \approx \Gamma_0 = \varphi(1,0,0) = \vartheta(\Omega^2)\)

\(g(1;0) > g(1,0_{[g(0)]}) \approx svo = \vartheta(\Omega^\omega)\)

\(g(1;0_{[g(0)]}) \approx \vartheta(\Omega^\omega\omega)\)

TREE(n) function \(≥ f_{\vartheta(\Omega^\omega\omega)}(n)\)

\(g_1(0) > g(1;0_{[g(1;0)]}) \approx \vartheta(\Omega^{\omega+1})\)

\(g_1(1) \approx \vartheta(\Omega^{\omega+2})\)

\(g_1^2(0) > g_1(g_1(0)) \approx \vartheta(\Omega^{\omega.2})\)

\(g_1(1,0) \approx \vartheta(\Omega^{\omega.3})\)

\(g_1(1;0) \approx \vartheta(\Omega^{\omega^2})\)

\(g_1(1;0_{[2]}) = g_1(1;0;0) \approx \vartheta(\Omega^{\omega^3})\)

\(g_1(1;0_{[g(0)]}) \approx \vartheta(\Omega^{\omega^{\omega}})\)

\(g_2(0) \approx \vartheta(\Omega^{\omega^{\omega^{\omega}}}) = \vartheta(\Omega^{\omega\uparrow\uparrow 3})\)

\(g_3(0) \approx \vartheta(\Omega^{\omega\uparrow\uparrow 4})\)

\(g_{g(0)}(0) \approx \vartheta(\Omega^{\omega\uparrow\uparrow\omega}) = \vartheta(\Omega^{\varphi(1,0)})\)

\(g_{S(g(0),1,1)}(0) \approx \vartheta(\Omega^{\varphi(1,1)})\)

\(g_{S(g(0),2,1)}(0) \approx \vartheta(\Omega^{\varphi(1,\omega^2)})\)

\(g_{S(g(0),3,1)}(0) \approx \vartheta(\Omega^{\varphi(1,\varphi(1,0))}) = \vartheta(\Omega^{\varphi^2(1,0_*)})\)

\(g_{S(g(0),g(0),1)}(0) \approx \vartheta(\Omega^{\varphi(2,0)})\)

\(g_{g(1)}(0) \approx \vartheta(\Omega^{\varphi(1,0,0)})\)

\(g_{g(1;0)}(0) \approx \vartheta(\Omega^{\Omega})\)

Large Veblen ordinal \(LVO ≥ f_{\vartheta(\Omega^\Omega)}(n)\)

\(g_{g(2;0)}(0) \approx \vartheta(\Omega^{\Omega^2})\)

Bird's H(n) function \(\approx f_{\vartheta(\varepsilon_{\Omega+1})}(n) = f_{\vartheta(\Omega\uparrow\uparrow\omega)}(n)\)

Further References
Further references to relevant blogs can be found here: User:B1mb0w