User blog:B1mb0w/The Hyper Function

The Hyper Function
This blog will compare recursive functions such as Veblen to my new Hyper function.

Notation Explained
The notation I use here is not in general use, but I find helpful. They are parameter subscript brackets, leading zeros assumption, recursion parameter subscript \(*\), and the decremented function \(C\).

Parameter Subscript Brackets, where:

\(M(a,0_{[2]}) = M(a,0,0)\)

\(M(a,0_{[2]},b_{[3]},1) = M(a,0,0,b_1,b_2,b_3,1)\)

Leading Zeros Assumption, where:

\(M(0_{[x]},0_{[2]},b_{[3]},1) = M(b_1,b_2,b_3,1)\)

Recursion Parameter Subscript \(*\), where:

\(M^2(a) = M^2(a_*) = M(M(a))\) and \(M(a,b_*) = M(a,b)\)

\(M^2(a,b_*) = M(a,M(a,b))\)

\(M^2(a_*,b) = M(M(a,b),b)\)

Decremented Function \(C\), where for any function:

\(M(a_{[b]},c + 1,d_{[e]})\) then \(C = M(a_{[b]},c_*,d_{[e]})\)

where parameter \(c\) is assumed to have the recursion parameter subscript \(*\).

Comparing Veblen, Hyper and other functions
For the comparison below, I will take some liberties with the definition of the Veblen Function. A separate comparison of Veblen function my Big number function is available here.

The Big number function behaves like the FGH function up to a point. The g function acts like a stronger version of the Veblen function as show here:

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

\(B(g(0),n) = f_{\omega}(n) = f_{\varphi(1)}(n)\)

\(B(g_c(a_{[b]}),n) >= B(g(a_{[b]}),n) > f_{\varphi(a_{[b]})}(n)\) for any values of \(a, b, c\)

Growth rate of the Big number function
The growth rate of the Big number function is compared here against the FGH using Veblen function and collapsing ordinal function.

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