# Feesh number1

FEESSHHH fish fish fish

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 fω^2(x), which is fast-growing hierarchy and I’ll talk about it later.

Define S1(x, y) to be Ackermann’s function, namely:

• S1(0, y) = y + 1
• S1(x, 0) = S1(x – 1, 1)
• S1(x, y) = S1(x – 1, S1(x, y – 1))

Then define S2 as a function that uses S1 as its base, and Sn uses Sn-1 as its base. Sn is:

• S1(0, y) = y + 1
• Sz(0, y) = Sz-1(y, y) for z > 1
• Sz(x, 0) = Sz(x – 1, 1)
• Sz(x, y) = Sz(x – 1, Sz(x, y – 1))

The fish function F1(x) = Sx(x, x) grows about as fast as fω^2(x), and the function is equivalent to Taro’s multivariable Ackermann function, or Sz(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 = Sf(m)
• S2 : [m, f(x)] → [n, g(x)]

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

WOW!

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