User:Ynought

German guy

\(S_1=1\)

\(S_{n+1}=1+\frac{1}{S_n}\)

\(\text{Lim}_{n\to\infty}S_n=\varphi\)

I don't include \(0\) in \(\mathbb{N}\), but I define a new set \(\mathbb{N}_0\) as \(0\cup\mathbb{N}\), so it is \(\mathbb{N}\) with 0 included.

WeakSS
I define the set of Weak Sequences \(\mathbb{W}\) as: I abreviate a Weak Sequence \(S\) and a non-negative interger \(n\) as \(S[n]\), also let \(s_i\) denote the \(i\)-th entry of \(S\), then let \(l\) denote the amount of entries in \(S\)
 * 1) The empty sequence \(\)
 * 2) Any list of non-negative intergers of finite length

I define a computable map \(\begin{eqnarray*} \textrm{Expand} \colon \mathbb{W} \times \mathbb{N} & \to & \mathbb{W} \times \mathbb{N} \\ S[n] & \mapsto & \textrm{Expand}(S[n]) \end{eqnarray*}\) in the following way:
 * 1) If \(S=\), then:
 * 2) Put \(\text{Expand}(S[n])=[n]\)
 * 3) If \(S\neq\), then:
 * 4) If \(s_l=0\), then:
 * 5) Put \(\text{Expand}(S[n])=(s_1,s_2,s_3...s_{l-1})[n+1]\)
 * 6) If \(s_l\neq0\), then:
 * 7) Define \(I^k(A,c)=A\), where \(A\in\mathbb{W}\) and \(k,c\in\mathbb{N}\), as \(A\) with \(k\) entries of \(c\) appended to \(A\)
 * 8) Put \(\text{Expand}(S[n])=I^n(A,s_l-1)[n]\), where \(A\) is \(S\) but with \(s_l\) removed

Put \(\text{Lim}_{\text{WeakSS}}(n)=(n)[n]\) where \(n\in\mathbb{N}\)

Analysis
Since \(\text{WeakSS}\) is basically a list-interpretation of the FGH we have \((s_1,s_2,s_3...s_l)[n]=f_{s_1}(f_{s_2}(f_{s_3}(...f_{s_l}(n))))\), so \(\text{Lim}_{\text{WeakSS}}(n)=f_\omega(n)\)

Nested Sequence System
The Idea is to make things like \((1,4,7,(2,5,6,(3),2),8,6)\) be things that are full standart expressions. It seems like this will involve some sort of ordinal subtraction, but maybe I can avoid that.

Analysis will be done in a google sheet whose link I will put here when I define this.

Definition Prototype
Define the set of Nest Sequences \(\mathcal{N}\) and a helper set \(\mathcal{H}\) simeltaniously in the following way:
 * 1) Let \(\in\mathcal{N}\)
 * 2) For all \((S)\in\mathcal{N}\) let \((S)\in\mathcal{H}\)
 * 3) For all \((S)\in\mathcal{N}\) and \(n\in\mathbb{N}\) let \((S,n)\in\mathcal{N}\)
 * 4) For all \((S)\in\mathcal{N}\) and \((A)\in\mathcal{H}\) let \((S,(A))\in\mathcal{N}\)

Define the set of Primitive Nest sequences \(\mathcal{N}\cap\text{Pr}\) in the following way:
 * 1) Let \(\in\mathcal{N}\cap\text{Pr}\)
 * 2) For \((A)\in\mathcal{N}\cap\text{Pr}\) and \(n\in\mathbb{N}\) let \((A,n)\in\mathcal{N}\cap\text{Pr}\)

Define a computable map \(ddd\) in the following way:

Mega Sequence System
- Expand \((A)[b]\) as a \((\text{Slightly less than }A)[b]\)-Sequence could possibly do A as <A[b] where <A gets created by a <<A[b]...

I don't think I will be able to define this, but it will be really strong if I make it.

Analysis will be done in a google sheet whose link I will put here when I define this.

Definition Prototype (Don't worry if there's nothing here that's to be expected)
Define the set of Mega Sequences \(\mathbb{M}\) in the following way:
 * 1) \(\in\mathbb{M}\)
 * 2) Let all sequences of non-negative integers of finite length be in \(\mathbb{M}\)

HMS
Hyper Matrix system

Allow things like (0,0,0,0)(3,3,3,3)(4,4,4,2)

Could be interesting how upgrading works here cuz it could possibly be pretty strong

Analysis will be done in a google sheet whose link I will put here when I define this.

Definition Prototype
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