Moved to Zeta-0/Revamp
Definition[]
Zeta-naught is a small countable ordinal, defined as the first (or least) fixed point of the epsilon-function.
It is usually denoted as ζ(0). The choice of Zeta, is because it is the next letter in the greek alphabet after epsilon. Likewise, the Zeta-function is the next normal function in the sequence of normal functions beginning with the epsilon function.
Technical Details[]
The Epsilon Numbers, ε0, ε1, ε2, are the fixed points of the ordinal function φ(α) = ω^α. It might be assumed that any power of omega will always be greater than power. ie.
ω^2 > 2 , ω^ω > ω , etc.
However if we define the supremum of the sequence ω , ω^ω , ω^ω^ω , ... etc. we can imagine obtaining an infinite power tower of the form:
ω^ω^ω^ ...
If we then compute φ(ω^ω^ω^...) we obtain ω^ω^ω^ω^...
This however is not any taller than the original power tower (because it was already infinite). Thus we have an ordinal that is a fixed point of φ.
Cantor denoted this ordinal as ε0. The main property of epsilon-0, and epsilon numbers in general is ω^ε(α).
To obtain new epsilon numbers from previous epsilon numbers one can obtain ε(α+1) with the sequence ω^(ε(α)+1) , ω^ω^(ε(α)+1) , ω^ω^ω^(ε(α)+1) , etc. For limit ordinals one can use its fundamental sequence. Let ε(λ) is the supremum of ε(λ[0]) , ε(λ[1]) , ε(λ[2]) , ... etc.
This allows us to define Epsilon numbers up to any ordinal α. We then define ζ(0) as the supremum of the sequence ε(0) , ε(ε(0)) , ε(ε(ε(0))) , ... etc.
It can be thought of informally as an infinitely nested epsilon function: ε(ε(ε(ε(ε( ... ))))). The main property of ζ(0) is that ε(ζ(0)) = ζ(0). It can be thought of as "so large" that applying the Epsilon Function does not make it any larger. We can however create larger ordinals by first taking the successor: ζ(0)+1. Note that Every Zeta Number is also an Epsilon Number. This means it is also true that ω^ζ(0) = ζ(0). However we can still get a larger ordinals from these functions if we plug in the successor instead: ω^(ζ(0)+1) > ζ(0), ε(ζ(0)+1) > ζ(0).
Within Fictional Googology[]
Zeta-naught is a transfinite ordinal from real mathematics, specifically an area of mathematics called set theory. It is used to discuss ordered sets.
It is a popular choice of ordinal to include within 0to Videos. Within Fictional Googology it is recognized as a "Tienum-Class Number". In fact every Transfinite Number from Set Theory is what forms the Tienum-Class Numbers. No distinction is made between Transfinite Ordinals and Transfinite Cardinals, thus it is not uncommon for them to be mixed and ranked together.
ζ(ζ(0))
ζ(ζ(ζ(0)))
ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(0))))))))
ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(ζ(0)))))))))))))))))))))))
Eta-0