User blog:B1mb0w/The Rex Function

The Rex Function
The Rex function generates very large numbers. It has a growth rate \(\approx f_{LVO}(n)\).

The Rex Function is a family of functions \(R\), \(e\) and \(r\) which use this simple rule set:

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

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

\(R(x(0), n) = R(n,n)\) and other instances of \(n\) can be substituted with \(x(0)\)

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

and

\(x(1, 0) = x^{x(0)}(0)\)

\(x(1, a + 1) = x^{x(1, a)}(x(1, a))\)

\(x(b + 1, 0) = x^{x(b, 0)}(b, 0_*)\)

\(x(1, 0, 0) = x^{x(1, 0)}(1_*, 0)\)

and

\(e(0) = x(1, 0_{[x(0)]})\)

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

\(e(1, 0) = e^{e(0)}(0)\)

\(e(1, a + 1) = e^{e(1, a)}(e(1, a))\)

\(e(b + 1, 0) = e^{e(b, 0)}(b, 0_*)\)

\(e(1, 0, 0) = e^{e(1, 0)}(1_*, 0)\)

and

\(R(1, 0, n) = R(e(1, 0_{[e(0)]}),n)\)

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:

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

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

\(R(R(1,x(0)),n) = f_{\omega.2}(n)\)

\(R(R(2,x(0)),n) = f_{\omega.2^{\omega}}(n)\)

\(R(R(3,x(0)),n) = f_{\varphi(1,0)}(n)\)

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

\(R(x(1),n) = R(R^{x(0)}(x(0)_*,x(0)),n) > R(R^{x(0)}(3_*,x(0)),n) \approx f_{\varphi(1,0,0)}(n) = f_{\Gamma_0}(n)\)

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

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

\(R(R(R(1,R^{x(0)}(3_*,x(0))),x(1)),n) > f_{\varphi(1,1,0)}(n)\)

\(R(R(R(\beta,r(1)),R(\beta,r(1))),n) > f_{\varphi(1,1,1)}(n)\)

Using the identity defined above

\(R(R(R(R(\beta,r(1))),r(1)),n) > f_{\varphi(1,1,1)}(n)\)

\(R(R^2(R(R(\beta,r(1))),r(1)_*),n) > f_{\varphi(1,1,2)}(n)\)

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

\(R(R^3(r(1)_*,r(1)),n) > f_{\varphi(1,3,0)}(n)\)

\(R(R^{r(0)}(r(1)_*,r(1)),n) > f_{\varphi(1,\omega,0)}(n)\)

\(R(R^{r(1)}(r(1)_*,r(1)),n) > f_{\varphi(1,\varphi(1,0,0),0)}(n) = f_{\varphi^2(1,0_*,0)}(n)\)

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

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 \(\approx f_{\vartheta(\varepsilon_{\Omega+1})}(n) = f_{\vartheta(\Omega\uparrow\uparrow\omega)}(n)\)

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

\(R^{b-1}(R(a,b),b_*) > R(R(a,b),b) > R(a,b)\)

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