Erdogdu ordinal notation

Erdogdu ordinal notation is a notation to show helix ordinals. The notation was proposed by Mert Erdoğdu to generalize the helix ordinals.
==Helix Ordinals==
There are three ordinals defined without the generalised Erdoğdu ordinal notation:
The original helix ordinal 𐰴0 defined as the length of a helix concentric with an apeirogon, wherein the apex is positioned at the origin and extends to a height of ω. The value of 𐰴 is ω√(1+2π) ≈ ω2.6987. 
The second helix ordinal or the large helix ordinal ("𐰚") 𐰴1 is defined as ω-th iterated integral of 𐰴0.The value of 𐰚 is (ω^ω)√(1+2π)/ω! ≈ exp(ω)/√(2πω) ≈ exp(ω)/10.
The third helix ordinal or the Erdoğdu ordinal ("𐰢") 𐰴2 is defined as ε0-th iterated integral of 𐰴0. The value of 𐰢 is (ω^ε0)√(1+2π)/ε0! ≈ exp(ε0)/√(2πε0) ≈ exp(ε0)/10.https://online.flippingbook.com/view/670059321/
==Erdoğdu ordinal notation==
The notation is defiened as follows:
Using the veblen function
a𐰴 = (ω^φ(a,0))√(1+2π)/φ(a,0)! ≈ exp(φ(a,0))/√(2πφ(a,0)) ≈ exp(φ(a,0))/10.
From the given fundamental sqeuence:
{ω√(1+2π),(ω^ω)√(1+2π)/ω!,(ω^ε0)√(1+2π)/ε0! ... }
== Limit of the notation ==
Let 𐰨 fixed point of iteratively using the notation given by the fundamental sequence:
{0, 0𐰴𐰴0,𐰴 ...}
{0, ω√(1+2π), (ω^φ(ω√(1+2π),0))√(1+2π)/φ(ω√(1+2π),0)!, (ω^φ((ω^φ(ω√(1+2π),0))√(1+2π)/φ(ω√(1+2π),0)!,0))√(1+2π)/φ((ω^φ(ω√(1+2π),0))√(1+2π)/φ(ω√(1+2π),0)!,0)!}
≈ {0, exp(ω)/10, exp(φ(exp(ω)/10,0))/10, exp(φ(exp(φ(exp(ω)/10,0)),0))/10, exp(φ(exp(φ(exp(φ(exp(ω)/10,0)),0))/10,0))/10}
> {0, exp(ω), exp(φ(exp(ω),0)), exp(φ(exp(φ(exp(ω),0)),0)), exp(φ(exp(φ(exp(φ(exp(ω),0)),0)),0))}
> {0, w, φ(ω,0), φ(φ(ω,0),0), φ(φ(φ(ω,0),0),0)}
≈ Γ0
𐰨 is the fixed point of the following:
𐰨[x+1] = (ω^φ(𐰨[x],0))√(1+2π)/φ(𐰨[x],0)! ≈ φ(exp(ε0)/√(2πε0),0) ≈ φ(exp(𐰨[x])/10,0)
== Sources ==
Category:Ordinal collapsing functions
Category:Ordinal notations
Category:Transfinite ordinals
Category:Transcendental numbers