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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

Fish showed in his book[1] (pp. 102-103) that S map and SS map are comparable to Taro's 4-variable Ackermann function; when we apply $$a$$ times of SS application and $$b$$ times of the initial S application to a function f(n)=n+1 (pair of a certain number and this function), the function obtained exactly matchs $$A(a,b,0,n)$$. As the definition of Fish number 2 applies SS map 63 times, and get a function is comparable to $$A(63,0,0,n)$$. Therefore Fish number 2 is comparable to $$A(1,0,0,0,63) \approx A(63, 0, 0, 63)$$.

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

Fish numbers: Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7
Mapping functions: S map · SS map · S(n) map · M(n) map · M(m,n) map
By Aeton: Okojo numbers · N-growing hierarchy
By BashicuHyudora: Primitive sequence number · Pair sequence number · Bashicu matrix system
By Kanrokoti: KumaKuma ψ function
By 巨大数大好きbot: Flan numbers
By Jason: Irrational arrow notation · δOCF · δφ · ε function
By mrna: 段階配列表記 · 降下段階配列表記 · 多変数段階配列表記 · SSAN · S-σ
By Nayuta Ito: N primitive
By p進大好きbot: Large Number Garden Number
By Yukito: Hyper primitive sequence system · Y sequence · YY sequence · Y function
Indian counting system: Lakh · Crore · Tallakshana · Uppala · Dvajagravati · Paduma · Mahakathana · Asankhyeya · Dvajagranisamani · Vahanaprajnapti · Inga · Kuruta · Sarvanikshepa · Agrasara · Uttaraparamanurajahpravesa · Avatamsaka Sutra · Nirabhilapya nirabhilapya parivarta
Chinese, Japanese and Korean counting system: Wan · Yi · Zhao · Jing · Gai · Zi · Rang · Gou · Jian · Zheng · Zai · Ji · Gougasha · Asougi · Nayuta · Fukashigi · Muryoutaisuu
Other: Taro's multivariable Ackermann function · TR function · Arai's $$\psi$$ · Sushi Kokuu Hen

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