Functions

== My function 1 ==
(x -> n)1 = xx->(n-1), (x -> n)m = x{m+1}(x->n)m-1
Series:
(a -> b)_{c} = a{c+1}b
(x -> x)_{(x -> x)_{x}} = x{x{x+1}x+1}x
(x -> x)[x] ~ x{{1}}x
=== Numbers: a{c}b to {a,c(1)2} ===
[Linear notation]
\begin{array}{lcl} (a\rightarrow b)_{c}>a\{c+1\}b \\ (a\rightarrow b)_{(a\rightarrow b)_{c}} > a\{a\{c+1\}b+1\}b \\ (a\rightarrow b)[c] > a\{\{1\}\}c \\ (a\rightarrow b)[c,2] > a\{\{2\}\}c \\ (a\rightarrow a)[b,c] > a\{\{c\}\}b \\ (a\rightarrow a)[a,b,c] > a\{\{\{c\}\}\}b \\ (a\rightarrow a)[b,c[2]] > \{a,c,1,1,2\} \\ (a\rightarrow a)[b,c,2[2]] > \{a,c,2,1,2\} \\ (a\rightarrow a)[a,b,c[2]] > \{a,b,c,1,2\} \\ (a\rightarrow a)[a,a,b,c[2]] > \{a,a,b,c,2\} \\ (a\rightarrow a)[a,a,a,b,c[2]] > \{a,a,a,b,c\}\end{array}
=== Numbers: {a,c(1)2} to {a,c(1)(1)2} ===
[Dimensional notation]
\begin{array}{lcl} (a\rightarrow a)[b,c[3]] > \{a,c^{1/2}(1)2\} \\ (a\rightarrow a)[b,c[4]] > \{a,c^{1/2}(1)3\} \\ (a\rightarrow a)[a,b[c]] > \{a,b^{1/2}(1)c-1\} \\ (a\rightarrow a)[a,b[c,2]] > \{a,b(1),c-1,2\} \\ (a\rightarrow a)[a,a[b,c[2]]] > \{a,c(1)(1)2\}\end{array}
=== Numbers: {a,c(1)(1)2} to {a,b((1)c)2} ===
[Expanded dimensional notation]
\begin{array}{lcl} (a\rightarrow a)[b,c[2]]_{2} > \{a,c(2)2\} \\ (a\rightarrow a)[b,c[2]]_{3} > \{a,c(3)2\} \\ (a\rightarrow a)[a,b[2]]_{c} > \{a,b(c)2\} \\ (a\rightarrow a)[a,b[2]]_{c,2} > \{a,b(c,0)2\} \\ (a\rightarrow a)[a,b[2]]_{c,c,2} > \{a,b(c,c,0)2\}\end{array}
=== Numbers: {a,b((1)c)2} to X^^X&c ===
[Hyper notation]
\begin{array}{lcl} (a\rightarrow a)[a,a[2]]_{b,c[2]} > \{a,a((1)c)2\} \\ (a\rightarrow a)[a,a[2]]_{b,c,2[2]} > \{a,a((1)c,0)2\} \\ (a\rightarrow a)[a,a[2]]_{a,b,c[2]} > \{a,a((1)b,c)2\} \\ (a\rightarrow a)[a,a[2]]_{a,a,b,c[2]} > \{a,a((1)a,b,c)2\} \\ (a\rightarrow a)[a,a[2]]_{b,c[3]} > \{a,a((2)c-1)2\} \\(a\rightarrow a)[a,a[2]]_{b,c[4]} > \{a,a((3)c-1)2\} \\ (a\rightarrow a)[a,a[2]]_{a,b[c]} > \{a,a((c-1)b-1)2\} \\ (a\rightarrow a)[a,a[2]]_{a,b[c,2]} > \{a,a((c,0)b-1)2\} \\ (a\rightarrow a)[a,a[2]]_{a,b[c,c[c,2]]} > \{a,a(((c-1)c-1)b-1)2\}\end{array}
=== Numbers: X^^X&c to {a,c/2} ===
[Expanded hyper notation] (until (a -> b)!c), [Mega notation]
\begin{array}{lcl}(a\rightarrow a)[a,a[2]]_{[b,c[2]]_2} > X\uparrow\uparrow X \And c \\ (a\rightarrow a)[a,a[2]]_{[b,c[2]]_3} > X\uparrow\uparrow X \times 2 \And c \\ (a\rightarrow a)[a,a[2]]_{[a,b[2]]_c} > X\uparrow\uparrow X \times X \And c \\ (a\rightarrow a)[a,a[2]]_{[a,a[2]]_{b,c}} > X\uparrow\uparrow X \times X + X \And c \\ (a\rightarrow a)[a,a[2]]_{[a,a[2]]_{b,c[2]}} > X\uparrow\uparrow X \times X + (X\times2) \And c \\ (a\rightarrow a)[a,a[2]]_{[a,a[2]]_{a,b[c]}} > (X\uparrow\uparrow X)^X \And c \\ (a\rightarrow b)!c > X\uparrow^3 X \And c \\ (a\rightarrow b)!c!c > \{X,X,4\}\And c \\ (a\rightarrow b)!!c > \{X,X,X\}\And c \\ (a\rightarrow b)!!!c > \{X,X,X,X\}\And c \\ (a\rightarrow b)![c]c > \{X,X(1)2\}\And c \\ (a\rightarrow b)![c![c]c]c > \{X,X(1)2\}\And c\And c\end{array}
Definition:
(a -> b)!c = (a -> b)[a,b[2]]_[a,b[2]]_…_[a,b[2]] by c nests
(a -> b)!!c = (a -> b)!b!b!b!…!b!b!b by c nests
(a -> b)![c]c = (a -> b)!!!…!!!c by c nests
=== Numbers: {a,c/2} to ??? ===
Definition:
(a -> b)!_{2}c = (a -> b)![c![c!…[c![c]c]…c]c]c]c by c nests
(a -> b)!_{c}c = (a -> b)!_{c-1}[c!_{c-1}…[c!_{c-1}[c]c]…c]c]c by c nests
(a -> b)?c = (a ->_{2} c) = (a -> b)!_{c!_{…c!_{c!c}c…}c}c by c nests
(x ->_{0}x) = x^^x = (x -> x)
(x ->_{1}x) = (x -> x)!x
(x ->_{2}x) = (x -> x)?x
(x ->[x]x) = (x ->_{x ->_{…}x}x) by x nests
(x ->/2[x]x) = (x ->[x ->[x ->[…]x]x]x) by x nests
(x ->/x[x]x)
(x ->/(x->/x[x]x)[x]x)
(x -> x & 2)
(x -> x & x-> x)
(x -> x & x -> x & x -> x)
(x -> && x)
(x -> &&& x)
(x -> &[x] x)
A(n) = (n ->_{n}n)
The number = AGoogol(Googol)
== My function 2 ==
In 008 (Added in 13:00 UST)
n? = FOOT^n(10^100)
It said to be I stole the function, But is changed.
10? = BIG FOOT
100? > BIG FOOT
10?? = BIG FOOTPLEX
10???…??? = Lakemaka
x(?,2) = 10???…??? by x nests
x(?,x) = x(???…???,x-1) by x nests
x(?,?) = x(?,x)
x(?,?,y) = x(?,(x-1)(?,?,y),y-1)
If x? = x+1:
x? = x+1
x?? = x+2
x(?,2) = 2x
x(??,2) = 4x
x(?,3) = x2^x
x(?,x) ~ f[w](x)
x(? & 2) ~ f[w^w](x)
x(? & ?) ~ f[w^(w+1)](x)
x(? & ?,2) ~ f[w^(2w)](x)
x(? & ?,?) ~ f[w^(w^2)](x)
x(? & ? & ?) ~ f[w^^3](x)
x(? (&#?) 2) ~ f[w^^w](x)
x(? (&#?) ?) ~ f[w^^w+w](x)
x(? (&#??) 2) ~ f[2*w^^w](x)
x(? (&#?,2) 2) ~ f[w*w^^w](x)
x(? (&#?,?) 2) ~ f[w^w*w^^w](x)
x(? (&#?&2) 2) ~ f[(w^^w)^2](x)
x(? (&#?&?) 2) ~ f[(w^^w)^w](x)
x(? (&#?&#?) 2) ~ f[e1](x)
x(? (&##?) 2) ~ f[ew](x)
x(? (&###?) 2) ~ f[ee0](x)
x(? && 2) ~ f[z0](x)
x(? (&^?) 2) ~ f[p(w,0)](x)
x(? (&^^?) 2) ~ f[p(p(w,0),0)](x)
x(? (&{?}?) 2) ~ f[G0](x)
x(? ({&,&,1,2}) 2) ~ f[p(w,0,0)](x)
x(? ({&,&(1)2}) 2) ~ f[Y(O^w)](x)
x(? &_2 2) ~ f[Y(O_w)](x + 1)
x(? &_? 2) ~ f[Y(O_O)](x + 1)
x(? &_&_? 2) ~ f[Y(O_O_O)](x + 1)
x(? & y) = x(?,?,?,…,?,?,? & y-1) by x nests
? sometimes called x+1 but sometimes FOOTx(Googol)
10(? & ?) = Fag
10(? & ? & ? & ? … ? & ? & ? &) by Fag nests = Fag(? (&#?) ?) = Super fag
10(? (&#?) ?) = Hyper fag
10(? && ?)=10(? (&##########?) ?) = Bag
10(? &&& ?)=Tag
10(? (&^?) ?)=Boag
Senas{x} is first super Uncomputable function which grows faster than any Uncomputable function
I have mistakes in some functions:
# Does n? equals to FOOT^n(n)? No
== My function 3 ==
Z(x)=x^2+1
z(x)=x^^(x+3*2x)
d(x)=<<<x^*2>>>
sx(y)=ffx(ω)(y)
s0(65) > G, ~ {3,65,1,2}
d(0)=1
d(1)=6
d(2)=39
d(3)=1546
d^5(10^100) is largest computable function which is larger than loader number
Fake issue in Fast growing hierarchy is in https://googology.fandom.com/wiki/Fast_growing_hierarchy#Warning. It includes wrong information due to no issues and it’s not ill-defined. And also it’s computable even before ω1CK. s3(n) is in tetrational arrays and grows fast as X^^X&n. sω(n) is in Pentational arrays and grows fast as X^^^X&n. This notation grows much faster than fast growing hierarchy. It can easily reach big numbers and never smaller than Graham’s number if n is 65.
s0(n) = fω+1(n)
s(α,n)=sα(n)
s(0,n)=f(w+1,n)=f(f(0,w),n)>{n,n,1,2}
s(1,n)=f(w2,n)=f(f(1,w),n)>{n,n,n,2}
s(2,n)=f(w*2^w,n)=f(f(2,w),n)>{n,n(1)n}
s(w,n)=f(φ(ω,0),n)>X^^^X&n
s(w+1,n)=f(Γ(0),n)>{X,X,X}&n
s(w+2,n)=f(φ(1,1,0),n)>{X,X,1,2}&n
s(w2,n)=f(φ(2,0,0),n)>{X,X,X,2}&n
s(w^2,n)=f(φ(ω,0,0),n)>{X,X,X,X}&n
s(w^w,n)=f(θ(Ω^ω),n)>{X,X(1)2}&n
s(w^w^w,n)=f(θ(Ω^Ω^ω),n)
s(ε(0),n)=f(θ(ε(Ω+1),n)
s(ζ(0),n)=f(θ(ζ(Ω+1),n)
s(φ(ω,0),n)=f(θ(φ(Ω+1,0)),n)
s(Γ(0),n)=f(θ(Ω(2)),n)
s(φ(ω,0,0),n)=f(θ(Ω(3)),n)
s(θ(Ω^ω),n)=f(θ(Ω(ω)),n)
SCG(13) ~ s(θ(Ω^ω),13)
s(θ(Ω^Ω),n)=f(θ(Ω(Ω)),n)
s(θ(ε(Ω+1)),n)=f(ψ(Ι),n)
f(w,x,x)=f(f(w,x),x)
f(w,x,x,…,x,x)=f(f(f(…(f(w,x),x)…),x),x)
f(x()w)=f(w,x,x,…,x,x)
t(x) = x+1
t(x,y) = t(t(x,y),y-1)
t(x,y,z) = t(x,t(x,y,z),z-1)
t(x,y(1)2) = t(x,x,x,…,x,t(x,x,x,…,x,x,x),x-1) by y nests = t(x,x,x,…,x,x,x) by y nests
t(x,y(1)z) = t(x,x,x,…,x,x,x(1)z-1) by y nests
t(x,x(1)x,y(1)2) = t(x,x(1)x,x,x,…,x,x,x) by y nests
t(x,y(1)(1)2) = t(x,x(1)x,x(1)x,x………x,x(1)x,x(1)x,x) by y nests
t(x,y(2)2) = t(x,x(1)(1)(1)……(1)(1)(1)2) by y nests
t(x,x(y)2) = t(x,y(1,2)2)
t(x,x(y)2) = t(x,x(y-1)(y-1)(y-1)………(y-1)(y-1)(y-1)2) by x nests
t(x,y(1,3)2) = t(x,x(y,2)2)
t(x,y(1,z)2) = t(x,y(1,1,2)2) = t(x,x(y,z-1)2)
t(x,y(1,1(1)2)2) = t(x,x(1,1,1,…,1,1,2)2) by y nests
t(x,y((1))2) = t(x,x(1,1(1,1(1,1(………(1,1(1,1(1)2)2)2)………)2)2)2)2) by y nests
t(x,y(1 \ 2)2) = t(x,x(((…(((1)))…)))2) by y nests
t(x,y(1 \ z)2) = t(x,x(y \ z-1)2)
t(x,y(1 \ 1,2)2) = t(x,x(1 \ y)2)
t(x,y(1 \ 1 \ 2)2)
t(x,y(1 \\ 2)2) = t(x,x(1 \ 1 \ 1 ………… 1 \ 1 \ 2)2) by y nests
t(x,y(1,2(1)2\2)2) = t(x,x(1 \\\…\\\ 2)2) by y nests
t(x,y(1,z(1)2\2)2) = t(x,x(y,z-1(1)2\2)2)
t(x,y(1,2(1)1,2\2)2) = t(x,x(1,2(1)y\2)2) = t(x,x(1,1,1,…,1,1,2(1)y-1\2)2) by x nests
t(x,y(1 \2 2)2) = t(x,x(1,2(1,2(1,2(………(1,2(1)2\2)………)2\2)2\2)2\2)2) by y nests
t(x,y(1 \1,22)2) = t(x,x(1 \y2)2) = t(x,x(1,2(1,2(1,2(………(1,2(1)2\y-12)………)2\y-12)2\y-12)2\y-12)2) by x nests
t(x,y(1 \1 \ 22)2)
t(x,y(1 - 2)2) = t(x,x(1 \ 1 \ 1 \ … 1 \ 1 \ 22 … 222)2) by y nests
Warning: functions I’ll defined sometimes.
Not all approximations confirmed to be true, it’s difficult to find the exact approximation by comparing Super fast growing hierarchy to fast growing hierarchy! I will fix them soon. But larger numbers like gamma ordinal is more difficult, so it changed soon.
== Calculators made by me ==
Tetration calculator - https://www.desmos.com/calculator/rmrhzkwe7o