User blog:B1mb0w/The Big number function

The Big number function
The Big number function is a very fast growing function. It's growth rate is well beyond \(f_{LVO}(n)\).

The Big number function is a pair of functions \(B\) and \(g\) which use this simple rule set:

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

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

\(B(g(0), n) = B(n,n)\) and other instances of \(n\) can be substituted with \(g(0)\)

\(g(c + 1) = g(0, c + 1) = B^{g(c)}(g(c)_*,g(c))\)

and

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

\(g(b + 1, c + 1) = g^{g(b + 1, c)}(x(b, c_*))\)

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

and

\(g = g_0\)

\(g_{a + 1}(0) = g_a(1, 0_{[g_a(0)]})\)

\(g_{a + 1}(b + 1) = g_a(1, 0_{[g_{a + 1}(b)]})\)

\(g_{a + 1}(b + 1, 0) = g^{g(b,0)}(b,0_*)\)

\(g(b + 1, c + 1) = g^{g(b + 1, c)}(x(b, c_*))\)

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

and

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

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

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

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

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

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

and

\(T(1, 0, n) = T(r(1, 0_{[r(0)]}),n)\)

\(T(a + 1, 0, n) = T(a,r(1, 0_{[r(0)]}),n)\)

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 T-Rex Function ... to \(\Gamma_0\)
The T-Rex Function behaves like the FGH function up to a point. Also refer to more detailed explanations in my previous blog The Rex Function:

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

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

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

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

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

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

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

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

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

\(T(T^2(x(1)),x(2)),n) > f_{\varphi(2,0,0)}(n)\)

\(T(x(3),n) \approx f_{\varphi(3,0,0)}(n)\)

\(T(x^2(0),n) \approx f_{\varphi(\omega,0,0)}(n)\)

\(T(x(1,0),n) \approx f_{\varphi(1,0,0,0)}(n)\)

\(T(e(0),n) = T(x(1,0_{[x(0)]}),n) = T(x(1,0_{[n]}),n) \approx f_{\varphi(1,0_{[n]})}(n) = f_{svo}(n)\)

Growth Rate ... to large Veblen ordinal (LVO) and beyond
The T-Rex Function is one of the Fastest Computable functions where:

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

\(T(3,x(0)) \approx \epsilon_0 = \varphi(1,0) = \vartheta(1)\)

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

\(e(0) \approx svo = \vartheta(\Omega^\omega)\)

\(e(1,0_{[x(0)]}) \approx \vartheta(\Omega^\omega\omega)\)

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

\(r(0) = e(1,0_{[e(0)]}) \approx \vartheta(\Omega^{\omega+1})\)

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

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

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

\(r(1,0_{[x(0)]}) \approx \vartheta(\Omega^{\omega^2})\)

\(r(1,0_{[e(0)]}) \approx \vartheta(\Omega^{\omega^{\omega}})\)

\(r(1,0_{[r(0)]}) \approx \vartheta(\Omega^{\omega\uparrow\uparrow\omega}) = \vartheta(\Omega^{\varphi(1,0)})\)

or

\(T(1,0,n) = T(r(1,0_{[r(0)]}),n) \approx f_{\vartheta(\Omega^{\varphi(1,0)})}(n)\)

\(T(x(0),0,n) \approx f_{\vartheta(\Omega^{\varphi(1,0)})}(n)\)

\(T(x(1),0,n) \approx f_{\vartheta(\Omega^{\varphi(1,0,0)})}(n)\)

\(T(e(0),0,n) \approx f_{\vartheta(\Omega^{\Omega})}(n)\)

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

\(T(e(1),0,n) \approx f_{\vartheta(\Omega^{\Omega^2})}(n)\)

\(T(e(2),0,n) \approx f_{\vartheta(\Omega^{\Omega^3})}(n)\)

\(T(e^2(0),0,n) \approx f_{\vartheta(\Omega^{\Omega^{\omega}})}(n)\)

\(T(e(1,0),0,n) \approx f_{\vartheta(\Omega\uparrow\uparrow 3)}(n)\)

\(T(e(1,0,0),0,n) \approx f_{\vartheta(\Omega\uparrow\uparrow 4)}(n)\)

\(T(e(1,0_{[x(0)]}),0,n) \approx f_{\vartheta(\Omega\uparrow\uparrow\omega)}(n)\)

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

\(T(r(0),0,n) = T(e(1,0_{[e(0)]}),0,n) \approx f_{\vartheta(\varepsilon_{\Omega+2})}(n)\)

\(T(r(1),0,n) \approx f_{\vartheta(\varepsilon_{\Omega.2})}(n)\)

\(T(r^2(0),0,n) \approx f_{\vartheta(\varepsilon_{\Omega^2})}(n)\)

\(T(r(1,0),0,n) \approx f_{\vartheta(\varepsilon_{\Omega^{\omega}})}(n)\)

\(T(r(1,0_{[x(0)]}),0,n) \approx f_{\vartheta(\varepsilon_{\Omega^{\omega^{\omega}}})}(n)\)

\(T(r(1,0_{[e(0)]}),0,n) \approx f_{\vartheta(\varepsilon_{\Omega^{\varphi(1,0)}})}(n)\)

\(T(1,0,0,n) = T(r(1,0_{[r(0)]}),0,n) \approx f_{\vartheta(\varepsilon_{\Omega^{\Omega}})}(n)\)

Bird's U(n) function \(\approx f_{\vartheta(\Omega_\omega)}(n)\)

Bird's S(n) function (new definition) \(\approx f_{\vartheta(\Omega_\Omega)}(n)\)

Some Identities
Some T-Rex Function identities are:

\(T(T(T(a,b)),b) > T(T(a,b),T(a,b))\)

because

\(T(T(T(a,b)),b) = T^b(T(a,b),b_*) = T(T(a,b),T^{b-1}(T(a,b),b_*))\)

and

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

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