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Fish number 2 (F2) is a number defined by Japanese googologist Fish in 2002.[1] Its base is the same as in Fish number 1:

• $$S_1(0,y) = y+1$$
• $$S_z(0,y) = S_{z-1}(y,y)$$ 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 function $$S_x(x,x)$$ grows about as fast as $$f_{\omega^2}(x)$$.

After that, we get a SS map. We have a function $$SS(3,x+1,S)$$ which expands to $$S_{x+1}(3,3)$$. Then we have an S map over that, which iterates the function $$x+1$$. If we have an SS map over that, we get a function with growth rate $$f_{\omega^22}(x)$$. Fish number 2 is defined as $$SS^{63}(3,x+1,S)$$

Original definition of Fish number 2 is actually slightly different from above and therefore the number is slightly different, although the order of the number approximated in FGH is similar. The definition is so complex and few people understood it when it was published. After discussions on BBS, the next version of Fish number, Fish number 3 was published, where the definition became simple and the number became larger. Therefore, after publication of $$F_3$$, $$F_1$$ and $$F_2$$ have only historical meaning.

## Approximations Edit

Notation Approximation
BEAF $$\{10,10,10,10,63\}$$
Hyper-E notation $$E100\#\#\#\#63$$
Chained arrow notation $$100 \rightarrow_{63} 100$$
Taro's multivariable Ackermann function $$A(1,0,0,0,63)$$
s(n) map $$s(4)[x+1](63)$$
m(n) map $$m(3)^3m(2)m(1)(63)$$
Fast-growing hierarchy $$f_{\omega^3}(63)$$
Hardy hierarchy $$H_{\omega^{\omega^3}}(63)$$
Slow-growing hierarchy $$g_{\varphi(63,0,0,0)}(100)$$

## Sources Edit

1. Fish, Googology in Japan - exploring large numbers (2013)