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Winner
This might become a Winner+ soon. BADGE: LOTSSSSS of potential and interesting things. |
A Brief Introduction[]
Welcome to the TheGoogologists power measurement list, a set of transcendentally boundless power levels which will be used to hopefully bring some legitimately high-quality hypergoogology and hypercosmology into this wiki. Because of it's increased quality, this page will be a lot longer than almost all other power measurements, factoring in various things such as mathematics, metaphysics, and even transfictionalism into the hierarchy itself. With that said, let us begin.
Countable Power Measurements[]
We shall now open with a proper definition of what a power level is. For the sake of brevity, let us assume for this page that a power level of 1 is equivalent to destroying a 1 meter three-dimensional cube of concrete, which is approximately 215,701,242,896,836,233,600 joules due to mass-energy equivalence. A power level of 2 would have twice the power of 1, rendering it equivalent to about 431,402,485,793,672,467,200 joules. This pattern continues onto the large finite numbers; as per tradition within this wiki we shall name some power levels corresponding to well-known googological constants:
- Googolpower - 10^100
- Googolplexpower - 10^10^100
- Grahampower - G(64) or {3, 64, 1, 2}
- TREEpower - TREE(3)
- Loaderpower - D5(99)
- Rayopower - Rayo(10^100)
- FOOTpower - FOOT10^100(10^100)
- LNGNpower - f10(10 ↑10 10)
The limit of all finite power levels is a power level of ω, the least transfinite ordinal. So far, it is equivalent to an infinite three-dimensional space, or in the VS Battles wiki's Tiering System High 3A, however through the power of fundamental sequences and ordinal collapsing functions we shall go much further with the early stages of this power hierarchy.
Since an infinite 3-dimensional space is equivalent to a finite 4-dimensional space, we can then assume that a power level of ω^2 is equivalent to affecting an infinite four-dimensional space, or in other words a Low 2C structure. In fact, for any finite n, a power level of ω^n will be equivalent to affecting an n+2-dimensional space. Eventually, we reach a power level of ω^ω, equivalent to affecting an infinite-dimensional space or a High 1B structure. We encounter a wall here, however, as the next tier, Low 1A, already corresponds to the least uncountable ordinal, ω1, when we have not even fully explored the limits of large countable ordinals yet. Knowing of a wall we will inevitably face soon, we will continue advancing up the hierarchy of countably infinite ordinals...
We may continue stacking ω^ω's on top of each other, until we reach an infinite power tower denoted as ω^ω^ω^ω^ω...; or in other words, the fixed point of ω^n where adding another ω into the power tower won't change the size of the new milestone we have just reached. To keep with standard mathematical discourse, we will name this fixed point ε0; also the point where a given ordinal power level catches up with the dimensional space it affects, i.e an ε0-level entity will be able to destroy an ε0-dimensional structure.
An infinite power tower of ε0 (ε0^ε0^ε0...) is denoted as ε1, and for all finite or infinite n, εn+1 is defined as an infinite power tower of εn. Eventually, nestings of ε0 begin to emerge, such as εε0 and εεε0, from which the limit of this sequence is ζ0, defined as an infinite nesting of ε's within each other. This pattern continues ad infinitum, however we only have a finite amount of Greek letters, so we will inevitably have to generalize this into an ordinal function known as the Veblen function if we want to advance any further.
The Veblen function is an ordinal function for creating large countable ordinals created by Oswald Veblen in 1908. It is denoted as φ(a, b, c...) where there can be any number of entries, and the entries themselves can assume any ordinal, even being able to nest Veblen functions onto themselves. A single-argument Veblen function, φ(n), is equivalent to ω^n, however we've already gone through this before. φ(n) is alternatively denoted as φ(0, n); with φ(1, 0) being equivalent to K = φ(K), or in other words, the ω^n fixed point or ε0. φ(2, 0) is the fixed point of φ(1, n) and is therefore equivalent to ζ0, therefore finally providing us with a way to diagonalize over the transfinite ordinals we have created earlier.
Eventually, we reach the fixed point of the binary Veblen function denoted as K = φ(K, 0), or in other words an infinite nesting of binary Veblen denoted as φ(φ(φ(...), 0), 0), known as the Feferman-Schutte Ordinal or simply Γ0. We will now introduce an extended Veblen function where Γ0 is equivalent to φ(1, 0, 0) and Γ1 is equivalent to φ(1, 0, 1). Eventually, the amount of arguments in the Veblen function itself begins to become large, eventually culminating in the Small Veblen Ordinal (SVO) which is the limit of all Veblen functions with a finite amount of arguments. Beyond this point, ordinal collapsing functions become the dominant method of creating large countable ordinals; however because there are simply too many OCFs of varying strengths and complexities we will simply skip over them for now.
We shall now name various power levels based on prominent large countable ordinals:
- Omegapower - ω
- Epsilonpower - ε0
- Zetapower - ζ0
- Binomegapower - φ(ω, 0)
- Gammapower - Γ0
- Ackerpower - φ(1, 0, 0, 0) (Ackermann ordinal)
- SVOpower - Ψ0(Ψ1(Ψ1(Ψ1(Ψ0(0))))) (Small Veblen ordinal with respect to Buchholz' psi function)
- LVOpower - Ψ0(Ψ1(Ψ1(Ψ1(Ψ1(0))))) (Large Veblen ordinal)
- BHOpower - Ψ0(Ψ2(0)) (Bachmann-Howard ordinal)