User blog:GamesFan2000/Ordinal Array Function

Here’s a notation based on the Psi Function that Lord Aspect made. I call this the Ordinal Array Function.

The first ordinal in the hierarchy is alpha, α. This ordinal is the base function of the entire system. It has a simple yet powerful effect. αn means to build an n-length array of n’s. Linear array notation already defeats chained arrows beyond four entries. This function will use BEAF’s rule set, since it’s the original array notation and linear arrays are well defined in it. α5 would be {5, 5, 5, 5, 5}. α20 is a twenty-entry array of twenties. You can express the function with a subscript or with brackets. From this point on, I’ll use brackets for convenience, i.e. α(n). Next up is α(n, a). This means to recurse the alpha function on n a times. α(4, 4)=α(α(α(α(4)))). α(n, a, b) means to recurse α(n, a) b times. α(4, 4, 4)=α(4, α(4, α(4, α(4, 4)))). Essentially, for the alpha function with any number of entries, the last entry tells you how many times to recurse the previous entries. The next level of the notation is α([n]). This becomes α(n, n, n, n, …n n’s…n). α([n, a]) becomes α([α([…a alphas…α([n])…])]). The process is the same, just with the square brackets this time, for any number of entries. α([[n] ])=α([n, n, n, n, …n n’s…n]). For an alpha function with any number of square brackets and one entry, it becomes an n-length array of n’s surrounded by one less square bracket, and for multi-entry alpha arrays you would recurse the previous entries in the function however many times the last entry asks for.

The next ordinal in the hierarchy is beta, β. β(n) means to recurse the alpha function with n square brackets n times on an n-length array of n’s. β(4)=α([[α([[[[α([[[[α([[[[4, 4, 4, 4] ])]]]])]]]])]]]]). β(n, a) takes a different approach. This time, the number of recursions on n is actually α(a). Yes, this is a huge step up from the previous level. β(4, 4) becomes β(β(β(β(…{4, 4, 4, 4} betas…β(4)…)))). β(n, a, b) tells you to recurse beta on (n, a) α(b) times, and so on and so forth. β([n]) means to create an α(n) length array of n’s within the beta function. β([4]) becomes β(4, 4, 4, 4, …{4, 4, 4, 4} 4’s…4). Other than the function using the alpha function for the recursions, the process for beta is exactly the same as the process for alpha. In fact, you can use this pattern for any ordinal. Let’s say that the next ordinal after beta is gamma, γ. γ(n) means to recurse the beta function with α(n) square brackets α(n) times on an α(n)-length array of n’s. γ(n, a) would then be β(a) recursions of gamma on n. Other than recursing it with the beta function, the process for moving through the levels is the same as before. Basically, for any ordinal, o(n), where o means the ordinal you’re using, means to recurse the ordinal the same way you would with the previous ordinal beyond the previous ordinal’s regular single variable function. For two variables or more and functions with square brackets, you use the previous ordinal and whatever the last variable in the function is to determine the recursion. In my hierarchy, you would go through all of the lowercase Greek letters from start to finish, or alpha to san, and then go through the uppercases, once again from alpha to san. (FYI, the complete Greek alphabet in order is alpha, beta, gamma, delta, epsilon, zeta, eta, theta, iota, kappa, lamda, mu, nu, xi, omicron, pi, rho, sigma, taut, upsilon, phi, chi, psi, omega, stigma, digamma, koppa, sampi, and san. At least, that’s the order that Word gives.) Following that, I’d go through the Coptic alphabet, again lowercase first and then uppercase. The order of the Coptic letters is shei, fei, khei, hori, gangia, shima, and dei. How you organize your hierarchy is up to you, as long as it follows the basic rules.