Fish number 1 is the smallest of the seven Fish numbers, created by Japanese googologist Fish. Fish number 1 is an extension to the Ackermann function, and the Fish function is comparable to f_{ω^2}(x), which is fast-growing hierarchy and I’ll talk about it later.

Define S_{1}(x, y) to be Ackermann’s function, namely:

S_{1}(0, y) = y + 1

S_{1}(x, 0) = S_{1}(x – 1, 1)

S_{1}(x, y) = S_{1}(x – 1, S_{1}(x, y – 1))

Then define S_{2} as a function that uses S_{1} as its base, and S_{n} uses S_{n-1} as its base. S_{n} is:

S_{1}(0, y) = y + 1

S_{z}(0, y) = S_{z-1}(y, y) for z > 1

S_{z}(x, 0) = S_{z}(x – 1, 1)

S_{z}(x, y) = S_{z}(x – 1, S_{z}(x, y – 1))

The fish function F_{1}(x) = S_{x}(x, x) grows about as fast as f_{ω^2}(x), and the function is equivalent to Taro’s multivariable Ackermann function, or S_{z}(x, y) = A(z, x, y).

To define Fish Number 1, we have to get complicated. Credits to Googology Wikia.

Define S map to be a map from a pair of number and function to a pair of number and function, or:

S : [m, f(x)] → [g(m), g(x)]

g(x) is:

B(0, n) = f(n)

B(m + 1, 0) = B(m, 1)

B(m + 1, n + 1) = B(m, B(m + 1, n))

g(x) = B(x, x)

Now define SS map to be a map from a set of number, function and S map to a set of number, function and S map. As follows:

SS: [m, f(x), S] → [n, g(x), S2]

S2, g(x) and n are:

S2 = S^{f(m)}

S2 : [m, f(x)] → [n, g(x)]

Apply SS map 63 times to [3, x + 1, S] to evaluate Fish number 1.