The R Function[]
THIS BLOG HAS BEEN REPLACED BY MY BLOG ON The Rex Function WHICH IS MUCH STRONGER.
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 use this simple ruleset:
\(R(n) = R(0,n) = n + 1\)
\(R(a + 1, n) = R^n(a,n_*)\)
\(R(r(0), n) = R(n,n)\) and other instances of \(n\) can be substituted with \(r(0)\)
\(r(a + 1) = R^{r(a)}(r(a)_*,r(a))\)
and
\(r(1, 0) = r^{r(0)}(0)\)
\(r(1, a + 1) = r^{r(1, a)}(r(1, a))\)
\(r(b + 1, 0) = r^{r(b, 0)}(b, 0_*)\)
\(r(1, 0, 0) = r^{r(1, 0)}(1_*, 0)\)
and
\(R(1, 0, n) = R(r(1, 0_{[r(0)]}),n)\)
Some Identities[]
Some R Function identities are:
\(R(R(R(a,b)),b) > R(R(a,b),R(a,b))\)
because
\(R(R(R(a,b)),b) = R^b(R(a,b),b_*) = R(R(a,b),R^{b-1}(R(a,b),b_*))\)
and
\(R^{b-1}(R(a,b),b_*) > R(R(a,b),b) > R(a,b)\)
Notation Explained[]
I use notation that is not in general use, but I find helpful. They are the \(*\) and parameter subscript brackets.
The \(*\) notation is used to explain nested functions. For example:
\(M(a) = M(a)\)
\(M^2(a) = M(M(a))\)
then let
\(M^2(a,b_*) = M(a,M(a,b))\)
\(M^2(a_*,b) = M(M(a,b),b)\)
Parameter subscript brackets are useful for functions with many parameters:
\(M(a) = M(a)\)
\(M(a,b) = M(a,b)\)
then let
\(M(a,0_{[1]}) = M(a,0)\)
\(M(a,0_{[3]}) = M(a,0,0,0)\)
\(M(a,b_{[2]}) = M(a,b_1,b_2)\)
\(M(a,0_{[2]},b_{[3]},1) = M(a,0,0,b_1,b_2,b_3,1)\)
Growth Rate of the R Function ... to \(\Gamma_0\)[]
The R Function behaves like the FGH function up to a point:
\(R^h(g,n_*) = f_g^h(n)\)
\(R(r(0),n) = f_{\omega}(n)\)
\(R(R(1,r(0)),n) = f_{\omega.2}(n)\)
\(R(R(2,r(0)),n) = f_{\omega.2^{\omega}}(n)\)
\(R(R(3,r(0)),n) = f_{\varphi(1,0)}(n)\)
\(R(R(r(0),r(0)),n) \approx f_{\varphi(\omega,0)}(n)\)
\(R(R(R^2(r(0)),r(0)),n) = f_{\varphi(\omega,0)}(n)\)
\(R(R(R(3,r(0)),r(0)),n) = R(R^2(3_*,r(0)),n) \approx f_{\varphi(\varphi(1,0),0)}(n) = f_{\varphi^2(1_*,0)}(n)\)
\(R(R^{r(0)}(3_*,r(0)),n) \approx f_{\varphi^n(1_*,0)}(n) = f_{\varphi(1,0,0)}(n)\)
\(R(r(1),n) = R(R^{r(0)}(r(0)_*,r(0)),n) > R(R^{r(0)}(3_*,r(0)),n) \approx f_{\varphi(1,0,0)}(n) = f_{\Gamma_0}(n)\)
Growth Rate ... to \(\varphi(1,0,0,0)\)[]
The R Function will eventually reach and surpass the small Veblen ordinal (svo):
\(R(R(r(0),r(1)),n) \approx f_{\varphi(\omega,\varphi(1,0,0)+1)}(n)\)
\(R(R(R(r(0),r(0)),r(1)),n) \approx f_{\varphi(\varphi(\omega,0),\varphi(1,0,0)+1)}(n)\)
\(R(R(R^{r(0)}(3_*,r(0)),r(1)),n) \approx f_{\varphi(1,0,1)}(n)\)
Let \(r(1) > \alpha = R^{r(0)}(3_*,r(0)) \approx \varphi(1,0,0)\)
\(R(R(\alpha,r(1)),n) \approx f_{\varphi(1,0,1)}(n)\)
\(R(R(1,R(\alpha,r(1))),n) \approx f_{\varphi(1,0,1).2}(n)\)
\(R(R(3,R(\alpha,r(1))),n) \approx f_{\varphi(1,\varphi(1,0,1)+1)}(n)\)
\(R(R(r(0),R(\alpha,r(1))),n) \approx f_{\varphi(\omega,\varphi(1,0,1)+1)}(n)\)
\(R(R(\alpha,R(\alpha,r(1))),n) \approx f_{\varphi^n(1_*,\varphi(1,0,1)+1)}(n) = f_{\varphi(1,0,2)}(n)\)
or
\(R(R^2(\alpha,r(1)_*),n) \approx f_{\varphi(1,0,2)}(n)\)
\(R(R^3(\alpha,r(1)_*),n) \approx f_{\varphi(1,0,3)}(n)\)
\(R(R^{r(0)}(\alpha,r(1)_*),n) \approx f_{\varphi(1,0,\omega)}(n)\)
\(R(R^{R(1,r(0))}(\alpha,r(1)_*),n) \approx f_{\varphi(1,0,\omega.2)}(n)\)
\(R(R^{R(2,r(0))}(\alpha,r(1)_*),n) > f_{\varphi(1,0,\omega^2)}(n)\)
\(R(R^{R(3,r(0))}(\alpha,r(1)_*),n) > f_{\varphi(1,0,\varphi(1,0))}(n)\)
\(R(R^{\alpha}(\alpha,r(1)_*),n) > f_{\varphi(1,0,\varphi(1,0,0))}(n)\)
or
\(R(R^{r(1)}(\alpha,r(1)_*),n) = R(R(\alpha),r(1)),n) > f_{\varphi^2(1,0,0_*)}(n)\)
\(R(R(R^2(\alpha),r(1)),n) > f_{\varphi^3(1,0,0_*)}(n)\)
\(R(R(R^{r(0)}(\alpha),r(1)),n) > f_{\varphi^n(1,0,0_*)}(n) = f_{\varphi(1,1,0)}(n)\)
or
\(R(R(R(1,\alpha),r(1)),n) > f_{\varphi(1,1,0)}(n)\)
Let \(\beta = R(1,\alpha)\)
\(R(R(\beta,r(1)),n) > f_{\varphi(1,1,0)}(n)\)
\(R(R(R(\beta,r(1)),R(\beta,r(1))),n) > f_{\varphi(1,1,1)}(n)\)
Using the identity defined above
\(R(R(R(R(\beta,r(1))),r(1)),n) > f_{\varphi(1,1,1)}(n)\)
\(R(R^2(R(R(\beta,r(1))),r(1)_*),n) > f_{\varphi(1,1,2)}(n)\)
\(R(R^{r(0)}(R(R(\beta,r(1))),r(1)_*),n) > f_{\varphi(1,1,\omega)}(n)\)
\(R(R^{r(1)}(R(R(\beta,r(1))),r(1)_*),n) > f_{\varphi(1,1,\varphi(1,0,0))}(n)\)
\(R(R(R^2(R(\beta,r(1))),r(1)),n) > f_{\varphi(1,1,\varphi(1,0,0))}(n)\)
\(R(R(R^{r(0)}(R(\beta,r(1))),r(1)),n) \approx f_{\varphi(1,1,\varphi(1,0,0).\omega)}(n)\)
\(R(R(R^{r(1)}(R(\beta,r(1))),r(1)),n) \approx f_{\varphi(1,1,\varphi(1,0,0)^2)}(n)\)
\(R(R(R^{R(\beta,r(1))}(R(\beta,r(1))),r(1)),n) \approx f_{\varphi^2(1,1,0_*)}(n)\)
\(R(R(R(1,R(\beta,r(1))),r(1)),n) \approx f_{\varphi^2(1,1,0_*)}(n)\)
\(R(R(R^2(1,R(\beta,r(1))_*),r(1)),n) \approx f_{\varphi^3(1,1,0_*)}(n)\)
\(R(R(R^{r(0)}(1,R(\beta,r(1))_*),r(1)),n) > f_{\varphi^n(1,1,0_*)}(n) = f_{\varphi(1,2,0)}(n)\)
or
\(R(R(R(2,R(\beta,r(1))),r(1)),n) > f_{\varphi(1,2,0)}(n)\)
or
\(R(R(R(2,R(R(1,\alpha),r(1))),r(1)),n) > f_{\varphi(1,2,0)}(n)\)
\(R(R(R(R(2,\alpha),r(1)),r(1)),n) > f_{\varphi(1,2,0)}(n)\)
\(R(R(R(r(1),r(1)),r(1)),n) > f_{\varphi(1,2,0)}(n)\)
\(R(R^2(r(1)_*,r(1)),n) > f_{\varphi(1,2,0)}(n)\)
\(R(R^3(r(1)_*,r(1)),n) > f_{\varphi(1,3,0)}(n)\)
\(R(R^{r(0)}(r(1)_*,r(1)),n) > f_{\varphi(1,\omega,0)}(n)\)
\(R(R^{r(1)}(r(1)_*,r(1)),n) > f_{\varphi(1,\varphi(1,0,0),0)}(n) = f_{\varphi^2(1,0_*,0)}(n)\)
\(R(r(2),n) > f_{\varphi^2(1,0_*,0)}(n)\)
then
\(R(R(r(1),r(2)),n) > f_{\varphi(1,\varphi(1,0,0) + 1,0)}(n)\)
\(R(R^{r(0)}(r(1)_*,r(2)),n) > f_{\varphi(1,\varphi(1,0,0) + \omega,0)}(n)\)
\(R(R(r(1)),r(2)),n) = R(R^{r(2)}(r(1)_*,r(2)),n) > f_{\varphi(1,\varphi^2(1,0_*,0),0)}(n) = f_{\varphi^3(1,0_*,0)}(n)\)
\(R^2(R(r(1)),r(2)_*),n) > f_{\varphi^4(1,0_*,0)}(n)\)
\(R^{r(0)}(R(r(1)),r(2)_*),n) > f_{\varphi^n(1,0_*,0)}(n) = f_{\varphi(2,0,0)}(n)\)
or
\(R(R^2(r(1)),r(2)),n) > f_{\varphi(2,0,0)}(n)\)
Let \(\gamma = R^2(r(1))\)
\(R(\gamma,r(2)),n) > f_{\varphi(2,0,0)}(n)\)
\(R(R(R(\gamma,r(2)),R(\gamma,r(2))),n) > f_{\varphi(2,0,1)}(n)\)
Using the identity defined above
\(R(R(R(R(\gamma,r(2))),r(2)),n) > f_{\varphi(2,0,1)}(n)\)
or
\(R(R(R(R(\gamma),r(2)),r(2)),n) > f_{\varphi(2,0,1)}(n)\)
\(R(R^2(R(\gamma)_*,r(2)),n) > f_{\varphi(2,0,1)}(n)\)
\(R(R^2(R^3(r(1))_*,r(2)),n) > f_{\varphi(2,0,1)}(n)\)
or
\(R(R^2(r(2)_*,r(2)),n) > f_{\varphi(2,0,1)}(n)\)
WORK IN PROGRESS
Growth Rate ... to svo[]
The R Function will eventually reach and surpass the small Veblen ordinal (svo):
WORK IN PROGRESS
The following is earlier work from my S Function.
\(S(n,g(1,0_{[g(0)]}),1) > S(n,g(1,0_{[n-1]}),1) \approx f_{\varphi(1,0_{[n]})}(n) = f_{svo}(n)\)
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