User blog:Flitri/An ordinal Collapsing up to the Least weakly Mahlo Cardinal

A Multivariate Ordinal notation: The Multistep function

1. Motivation:

Ordinal Collapsing Functions are most relevant to proof theory, set theory, ordinal analysis, as well as googology, a recreational mathematics term meaning the "the study of large numbers" after googol, a well known number known for being relatively large.

Here we give an multivariate ordinal collapsing function, hereafter abbreviated OCF, which starts & eventually generalizes Buchholz's Psi Function hierarchy. The major utility of this function comes from the fact that fundamental sequences are relatively easier to define for the function. The main purpose of this function is for it to produce large countable ordinals to be used in the Fast-Growing Hierarchy & related functional hierarchies. It is assumed the reader has knowledge on ordinal arithmetic, fixed points, & Cantor Normal Form of ordinals.

2. First Steps, up to the Bachmann-Howard Ordinal: Write Ord as the class of all ordinals & N as the set of natural numbers, also known as the finite ordinals. cof(x) = min{μ|α[μ] = x for some μ-indexed sequence α with α[δ] < α[λ] ⇌ δ < λ & δ, λ, μ in Ord} cof(0) = 0 & cof(n) = 1 & 0 < n < ω N = {x | x < ω} is the set of natural numbers Reg = {x | cof(x) = x & x > 1} is the set of infinite regular ordinals Lim = {x | cof(x) ≥ ω} is the set of limit ordinals Suc = {x | cof(x) = 1} is the set of successor ordinals ψ(α) is the smallest ordinal not constructible with elements of the set K(α) from the function set F: Formally: ψ(α) = min{ x | x not in K(α) over F} K[0](α) = {α} ⋃ {Ω} K[n+1](α) = K[n](α) ⋃ {f(x) | f in F & x K[n](α)} K(α) = ⋃[q in N]{K[q](α)} F = {i+1, i+j, i*j, ω^j, ψ(β) | i, j, β in K(α) & β < α} ω is the first Infinite Ordinal = min{ x | x > n & n in N} Ω is the first Uncountable Ordinal = min{ x | cof(x) > ω}
 * x| = the cardinal associated with the ordinal x

ψ(0) = ε(0) since ε(0) is the smallest ordinal satisfying x = ω^x, meaning it is equal to an infinite power tower of ω‘s.

ψ(1) = ε(0)^ε(0)^ε(0)^... = ε(1) ψ(ω) = Sup{ε(0), ψ(1), ψ(2), ...} = ε(ω)

Generally: ψ(α) = ε(α) when α < ζ(0), which is the smallest ordinal satisfying x = ε(x)

ψ(ζ(0)) = ζ(0) since ζ(0) satisfies x = ε(x)

ψ(ζ(0)+1) = ζ(0) because K(ζ(0)+1) already contains ζ(0) which cannot be constructed with finite applications of ε(x). This occurs ∀ζ(0) ≤ x ≤ Ω. To shorten notationabit we will write v < X — Y to signify that v runs from zero through all ordinals less than X then the expression involving v equals X as v runs through all ordinals inclusively between X & Y.

ψ(Ω) = ζ(0)

ψ(Ω+1) = ψ(Ω)^ψ(Ω)^ψ(Ω)^... = ε(ζ(0)+1)

ψ(Ω+α) = ε(ζ(0)+α) ∀α < ζ(1), the next ordinal satisfying x = ε(x)

ψ(Ω+ζ(1)) = ζ(1)

ψ(Ω+ζ(1)+1) equaling ζ(1) is in parallel to ψ(ζ(0)+1) being equal to ζ(0), which means that Ω is needed: ψ(Ω*2) = ζ(1)

Generally: ψ(Ω*(1+α)) = ζ(α) ∀α < η(0)—Ω & η(0) is the smallest ordinal satisfying x = ζ(x)

Note the f(1+α) > g(α) difference only occurs when α < ω. When α ≥ ω, they coincide: ψ(Ω*ω) = ζ(ω) since 1+ω = ω <  ω+1 (ordinal addition being non-communative)

Generally: ψ(Ω*(1+α)+β) = ε(ζ(α)+β) for α < η(0)—Ω & β < ζ(α+1)—Ω (fixing α)

ψ(Ω*η(0)) = η(0) is defined in parallel to ζ(0), Ω is needed similarly.

ψ(Ω^2) = η(0) & ψ((Ω^2)*2) = η(1)

More generally: ψ((Ω^α)*(1+)) = φ[1+α, β], with φ[x, y] being the 2-variable Veblen Function, an ordinal notation developed by Oswald Veblen for enumerated the fixed points of ω^x. φ[α+1, 0] is the smallest ordinal satisfying x = φ[α, x] & φ[0, x] = ω^x.

With the following bounds on α and β: 1 ≤ α < φ[1, 0, 0] — Ω & β < φ[α+1, 0] — Ω (fixing α)

ψ((Ω^7)*5) = φ[8, 4] & ψ((Ω^ω)*ω^3) = φ[ω, ω^3] are examples

The Supremum of this system is the Feferman-Schütte Ordinal, ψ(Ω^Ω) or Γ(0). It is the smallest ordinal satisfying x = φ[x, 0].

When we defined ψ(Ω^2) = η(0) the maximum structure using Ω changed from Ω to Ω^2, as opposed changing Ω to Ω*2 when going from ζ(0) to ζ(1), to go into η(1), namely ψ((Ω^2)*2) = η(1). These structures made with Ωs will be called Ω-structures, these will become important later on.

Some more values & families include:

ψ(Ω^Ω+α) = ε(Γ(0)+α) for α < ζ(Γ(0)+1)—Ω ψ(Ω^Ω+Ω*α) = ζ(Γ(0)+α) for α < η(Γ(0)+1)—Ω ψ(Ω^Ω+Ω^α) = φ[1+α,Γ(0)+1]

Generally: ψ(Ω^Ω+(Ω^α)*β) = φ[1+α, Γ(0)+β] b < φ[1+α, Γ(0)+1]—Ω (fixing α) & α < Γ(1)—Ω

ψ(Ω^Ω+(Ω^6)*9) = φ[7, Γ(0)+9] ψ(Ω^Ω+(Ω^ω)*5) = φ[ω, Γ(0)+5]

Eventually we reach the limit ψ((Ω^Ω)*Γ(1)) or ψ((Ω^Ω)*2), making Ω^Ω the largest Ω-structure currently in use.

ψ((Ω^Ω)*2) = Γ(1)

Generally: ψ((Ω^Ω)*(1+α)) = Γ(α) = φ[1, 0,α] for α < φ[1, 1, 0] — Ω

φ[1, 1, 0] is the smallest ordinal satisfying x = Γ(x) or x = φ[1, 0, x].

ψ((Ω^Ω)*ω^2) = φ[1, 0, ω^2] = Γ(ω^2) ψ((Ω^(Ω+1))*(1+α)) = φ[1, 1,α] for α < φ[1, 2, 0]—Ω

More generally: ψ((Ω^(Ω*α+β))*(1+γ)) = φ[α, β, γ], with the following bounds on α, β, γ:

1 ≤ α < φ[1, 0, 0, 0] — Ω & β < φ[1+α, 0, 0] — Ω (fixing α) & γ < φ[α,1+β, 0] — Ω (fixing α & β)

ψ((Ω^(Ω*8+5))*4) = φ[8, 5, 3] & ψ((Ω^(Ω*8+5))*ω) = φ[8, 5, ω] are examples

Finally we reach ψ(Ω^Ω^2) using Ω^Ω, the largest Ω-structure in use, also called the Ackermann ordinal or φ[1, 0, 0, 0] & the smallest ordinal satisfying x = φ[x, 0, 0].

3. Reaching higher Veblen arguments & the Bachmann-Howard Ordinal:

ψ(Ω^Ω^α) = the smallest ordinal satisfying x = φ[x, 0@α] with α zeros with α ≥ 1, since when α = 0 it collapses into ψ(Ω) = ζ(0) which satisfies x = φ[1, x].

ψ(Ω^Ω^ω) is the small Veblen Ordinal, which is the supremum of the generalized Veblen function with finitely many variables.

ψ(Ω^Ω^Ω) is the large Veblen Ordinal, which is the smallest ordinal satisfying x = φ[1, 0@x]

Eventually we reach ψ(ε(Ω+1)), the Bachmann-Howard Ordinal (BHO). It is limit of the notation & it is equal to Sup{ψ(Ω), ψ(Ω^Ω), ψ(Ω^Ω^Ω), ...}

4. Up to the omega fixed point:

To get past the Bachmann-Howard Ordinal, we must introduce a new ordinal Ω(1). this ordinal can just be any ordinal not producible by ψ(1, α) or an uncountable larger one. For the purposes of this notation, we shall choose it to be an uncountable ordinal, specifically the smallest regular uncountable ordinal whose cardinality is larger than Ω(0)’s cardinality. To make it shorter, to extend ψ(ν, α) past ψ(ν, ε(Ω(ν)+1)) & Ω(ν), we bring down a larger ordinal Ω(ν+1), & make ψ(ν+1, 0) = ε(Ω(ν)+1). L(ν) is the associated limit ordinal for the successor ordinal ν: L(N) → ω & L(ω+N) → ω*2 etc. The definition of ψ(ν, α) is as follows: ψ(ν, α) is the smallest ordinal not constructible with elements of the set K(ν, α) & the function set F with K(ν, α) being defined as follows: More formally: ψ(ν, α) = min{x | x not in K(ν, α)} K[0](ν, α) = {0} ⋃ {Ω(v)} K[n+1](ν, α) = K[n](ν, α) ⋃ {f(x, y), g(z) | f, g in F & x,y,z in K[n](ν, α)} K(ν, α) = ⋃[q in N]{K[q](ν, α)} Ω(α+1) = min{x ||x| = Ω(α+1)}—|X| is the associated cardinal of the ordinal X Ω(L) = Sup{Ω(k) | k < L} F = {i+1, i+j, i*j, ω^i, ψ(m, α), ψ(ν, c) | i, j, c, m in K(ν, α);m < ν;c < a} ψ(ν+1, α) = ε(Ω(ν)+1+α) for α < ζ(Ω(ν)+1)—Ω(ν+1) ψ(L, 0) = Sup{ψ(k, 0) | k < L} ψ(L, α) = ε(ψ(L, 0)+1+α) for 0 < α < ζ(ψ(L, 0)+1)—Ω(L) 0 < ψ(0, α) = ψ(α) <  Ω Ω(ν-1) <  ψ(ν, α)  <  Ω(ν)  <  Ω(L[ν]) & cof(ν) = 1 ψ(L, α) <  Ω(L) since ψ(k, α)  <  Ω(k) for k  <  L

Eventually we reach ψ(Ω, 0), the smallest ordinal satisfying x = ψ(x, 0) in the same fashion that ψ(Ω) is the smallest ordinal satisfying x = ψ(x). As a result, ψ(Ω, 0) starts a new Veblen Hierarchy started with φ[0, a] = ψ(α 0).

ψ(Ω, 0) = φ[1, 0] ψ(Ω, Ω(Ω)) = φ[2, 0] ψ(Ω, Ω(Ω)^^2) = φ[1, 0, 0]

we can continue on with ψ(Ω*2, 0) which is the smallest ordinal satisfying x = ψ(Ω+x, 0)

ψ(Ω^2, 0) = the smallest ordinal satisfying x = ψ(Ω*x, 0) & Ω-structure is Ω*Ω = Ω^2

ψ(Ω^Ω, 0) = the smallest ordinal satisfying x = ψ(Ω^x, 0) Ω-structure is Ω^Ω = Ω^^2

ψ(Ω^Ω^2, 0) = the smallest ordinal satisfying x = ψ(Ω^(Ω*x), 0) Ω-structure is Ω^(Ω*Ω)

ψ(Ω(Ω), 0) is the smallest ordinal satisfying x = ψ(Ω(x), 0) using the new function Ω(x) on ordinals x ≥ Ω.

Eventually K[ν](0) would contain {Ω, Ω(Ω), Ω(Ω(Ω)), ...} but not its fixed point. Normally in this scenario, one would introduce the first inaccessible cardinal I(0) & its respective axioms & write ψ(I, 0) as the fixed point of Ω(x), instead we will add a third argument to our function: ψ(α[1]| x, y).Notice the bar in the argument, that distinction becomes important later ψ(0[1]|) is the smallest ordinal satisfying x = Ω(x), also written as γ(0).

5. Past the omega fixed point & on to higher order Ordinals.

ψ(α[1]| β, γ) = min{z|z not in K(α, β, γ) over F} K[0](α, β, γ) = {α, β, γ} ⋃ {Ω[1]} K[n+1](α, β, γ) = K[n](α, β, γ) ⋃ {f(b) | f in F & b in K[n](α, β,γ)} K(α, β, γ) = ⋃[q in N]{K[q](α, β, γ)} F = {i+1, i+j, i*j, ω^j[β, γ], Ω(i)[α], ψ(δ[1]ξ, η) | δ,ξ,η,i,j in K(α x, y)};δ < α & β ≤ ξ & γ ≤ η}

The only discernible difference between ψ(ν,α[1]|) & ψ(ν, α) is their enumerated fixes points (ε(α) & γ(α)) respective structural ordinal hierarchies, which are Ω(ν) is ψ(ν, α)'s case & Ω[1](ν) is ψ(ν,α[1]| x, y)'s case.

ψ(v, α[1]| β, γ) = min{z|z not in K(v, α, β, γ) over F} K[0](v, α, β, γ) = {α, β, γ} ⋃ {Ω[1](v)} K[n+1](v, α, β, γ) = K[n](v, α, β, γ) ⋃ {f(b) | f in F & b in K[n](v, α, β,γ)} K(v, α, β, γ) = ⋃[q in N]{K[q](v, α, β, γ)} F = {i+1, i+j, i*j, ω^j[β, γ], Ω(i)[v, α], ψ(μ, δ[1]ξ, η) | δ,ξ,η,i,j in K(α x, y)};δ < α & β ≤ ξ & γ ≤ η & v ≤ μ} Ω[1](α+1) = {x | cof(x) > cof(Ω[1](α)) & x = Ω(x)} Ω[1](L) = Sup{Ω[1](k) | k < L}

ψ(0|0) = ε(γ(0)+1) The reason is that ψ(0|) is a well defined ordinal. It is the smallest fixed point of Ω(x). ψ(0|0) takes ψ(0|), sticks it into the fundamental set—making it {0, ψ(0|), Ω[1]}, & changing the function set to be {x+1, x+y, x*y, ω^x, Psi} so this is makes ψ(0|0) the next epsilon number after ψ(0|). ψ(0|1) = ε(γ(0)+2) is defined for a similar reason.

ψ(9|ω) = ε(γ(9)+ω) & Generally: ψ(α|b) = ε(γ(α)+1+b) for b < ζ(ψ(α|)+1)—X

But what is the structural ordinal (X) in this case? The answer is Ω(ψ(α|)+1). The additional +1 is so that the subscript is slightly larger than ψ(α|), since Ω(ψ(α|)) = ψ(α|). Ω(ψ(α|)+1) is the smallest Ω-number after ψ(α|), so it, due to it having a base of Ω, is a structural ordinal.

ψ(α|Ω(ψ(α|)+1)) = ζ(ψ(α|)+1) ψ(α|Ω(ψ(α|)+1)^Ω(ψ(α|)+1)) = Γ(ψ(α|)+1) ψ(α|1, 0) = ε(Ω(ψ(α|)+1)+1)

ψ(α[1]| x, 0) = ε(Ω(ψ(α|)+x)+1). Just how the 1+α in ψ(1+α, 0) increments α in ε(Ω(α)+1), the x in ψ(α[1]| x, 0) increments the Ω(ψ(α|)+c) in ε(Ω(ψ(α|)+c)+1). Note that x can be zero here. This occurs here since letting x = 0 collapses the function to ψ(α|0) = ε(ψ(α|)+1), & additionally ψ(|0, x) = ψ([1]|x)

To get smaller ordinals between Ω[1](z) & Ω[1](z+1) use ψ([z\1]| x, y) with the structural ordinal Ω(Ω[1](z)+1+x) & x, y ≥ 0

Ω[1] ≤ ψ([1]|α, β) < ψ(1, 0|) ∀α, β in Ord ψ([0\1]|) = Ω[1] ψ([0\1]|0) = ε(Ω[1]+1) ψ([0\1]|Ω(Ω[1]+1)^^2) = Γ(Ω[1]+1)

Reaching ψ(1, 0|) is the same as reaching ψ(1, 0) by exchanging γ(x) for ε(x): —————— Degenerate case 1: no η ψ[S](ν, α) = Enum{ x | x in Lim} < Ω(ν) ω ≤ cof(ψ[S](ν, α)) < Ω(ν) Structural Ordinal: Ω(ν) ——— ψ[S](ν, α) = min{ z | z not in K(ν, α)} K[0](ν, α) = {α} ⋃ {Ω(ν)} K[n+1](ν, α) = K[n](ν, α) ⋃ {f(x) | f in F & x in K[n](ν, α)} K(ν, α) = ⋃[q in N]{K[q](ν, α)} F = {β+1, ψ[S](ν, ξ) | β,δ,ξ in K[n](ν, α) & ν ≤ δ & ξ < α} —————— Degenerate case 2: no η ψ[Φ](α) = Enum{ x | x in Reg} < M ω ≤ cof(ψ[Φ](α)) < M Structural Ordinal: M ——— Let M is the least weakly Mahlo cardinal ψ[R](α) = min{ z | z not in K(α)} K[0](α) = {α} ⋃ {M} K[n+1](α) = K[n](α) ⋃ {f(x) | f in F & x in K[n](α)} K(α) = ⋃[q in N]{K[q](α)} F = {ψ[R](ξ),Δ[Φ,cof(δ) < δ](β)| β,ξ in K[n](α) & ξ < α} Δ[Φ,St](x) is any function Δ: Ord → Ord such that Δ(Φ,St](x) = α → Φ(St,α) = 1 for the statement St How Φ(St,x) works: Φ(St,x) → St(x) = True → 1 Φ(St,x) → St(x) = False → 0 —————— η = 0 ψ(β, α) = Enum{x | x = ω^x} < Ω(β) ω ≤ cof(ψ(β, α)) ≤ cof(Ω(β)) Ω(α) = Enum[min]{x | x = ω^x & |x| = ℵ(1+α) & x > ψ(α,λ) & λ in Ord} ψ(0, α) = ψ(α) Maximal Structure ordinal: Ω(β) —————- η = 1 ψ(δ, μ[1]|#) = Enum{x | x = Ω(x)} < Ω[1](δ) ω ≤ cof(ψ(δ, μ[1]|#)) ≤ cof(Ω[1](δ)) Ω[1](α) = Enum[min]{x | x = Ω(x) & x in Reg ⇌ cof(α) in {0,1} & x > ψ(α, λ[1]|#) & λ in Ord} ψ(0, μ[1]|#) = ψ(μ[1]|#) Bracket case: δ, μ—[δ]: ψ([δ\1]|#) < Ω[1](δ+1) Maximal Structure ordinal: Ω[1](δ) —— ψ(α, b[η]|) = min{z|z not in K(α, b)[η]} K[0](α, b)[η] = {α, b} ⋃ {Ω[η](a)} K[n+1](α, b)[η] = K[n](α, b)[η] ⋃ {f(x) | f in F & x in K[n](α, b)[η]} K(α, b)[η] = ⋃[q in N]{K[q](α, b)[η]} F {i+1, i+j, i*j, Ω[θ](i), ψ(d, c[θ]|) | i,j,α,b,c,d,θ in K[q](α, b)[η] & c < b, α ≤ d, η ≤ θ} —————— ψ(κ, π[η]|#) = Enum{x | x = Ω[θ](x)} < Ω[η](κ) ω ≤ cof(ψ(κ, π[η]|#)) ≤ cof(Ω[η](κ)) Ω[η](α) = Enum[min]{x | x = Ω[θ](x) & x in Reg ⇌ cof(α) in {0,1} & x > ψ(α, λ[η]|#) & λ in Ord} ψ(0, π[η]|#) = ψ(π[η]|#) Bracket case: κ, π—[κ\η] -p < η & ψ([κ\η]|δ, μ[p]) < Ω[η](κ+1) Maximal Structure ordinal: Ω[η](κ) —————— ψ(κ, π[1, 0|S]|) = Enum{x | x = F(x)} < Ω[1, 0|S](κ) S=0: F(x) = ψ(0[x]|) S=1: F(x) = Ω[x](0) ω ≤ cof(ψ(κ, π[1, 0|S]|#)) ≤ cof(Ω[1, 0|S](κ)) Ω[1, 0|S](α) = Enum[min]{x | x = F(x) & x in Reg ⇌ cof(α) in {0,1} & x > ψ(α, λ[1, 0|S]|#) & λ in Ord} ψ(0, π|#) = ψ(π|#) Bracket case: κ, π—[1, 0|S;κ] -p < F(p) & ψ([1, 0|S\κ]|δ, μ[p]) < Ω[1, 0|S](κ+1) Maximal Structure ordinal: Ω[1, 0|S](κ) ψ(κ, π[1, η|S]|) = Enum{x | x = F(x)} < Ω[1, η|S](κ) cof(η) in {0,1}, θ+1 = η & cof(η) ≥ ω, θ = {x | x < η} S=0: F(x) = ψ(0[1, θ, x |next(S)]|) S=1: F(x) = Ω[1, θ, x|next(S)](0) Each level is followed by a descending chain of signs indexed either by a subset of N (countable cofinality) or by the set {ξ | ξ < η} (uncountable cofinality) according to some sequence of signs S* or if not defined: Mx = {1,1, 1, ...} & Mn = {0, 0, 0, ...} based on the first sign. S* = {1, 0, 1, 0, ...} ψ(κ, π[2, 0 |1]|) = Enum{x | x = ψ(0[1, x|0]|)} < Ω[2, 0|1](κ) ψ(κ, π[1, 0, 0|0]|) = Enum{x | x = Ω[x, 0|1](0)} < Ω[1, 0, 0|1](κ) Limit is ψ[1](0) = min{ x | x = ψ(0[1, 0[0]x|M]|)} ψ[0](α,β[#]|#) = ψ(α,β[#]|#) —- α in Suc & β+1 = α & μ in Suc & θ+1 = μ & ξ in Lim ψ[α](γ,δ[0]|p) = Enum{ x | x = ψ[β](0[1, 0[0]x|M]|)} < Ω[α;0](γ,p) ψ[α](γ,δ[μ]|p) = Enum{ x | x = Ω[α;θ](x)} < Ω[α;μ](γ,p) ψ[α](γ,δ[ξ]|p) = Enum{ x | x = Ω[α;φ](x) & φ < ξ} < Ω[α;ξ](γ,p) —- α in Lim & μ in Suc & θ+1 = μ & ξ in Lim ψ[α](γ,δ[0]p|) = Enum[Sup]{ x | x = ψ[λ](0[1, 0@x|M]|) & λ < α} < Ω[α;0](γ,p) ψ[α](γ,δ[μ]p|) = Enum{ x | x = Ω[β;θ](x)} < Ω[α;μ](γ,p) ψ[α](γ,δ[ξ]p|) = Enum{ x | x = Ω[β;φ](x) & φ < ξ & β < α} < Ω[α;ξ](γ,p) —- ψ[1,0](α,β[γ]p|) → Enum{ x | x = Ω[x;γ](0)} < Ω[1,0;γ](α,p) [1,0,0;0] → [1;0;0] → [1;0[1]x] → [1:0] → [1:0[2]x] … Ω[#](γ,p) follow from the definition of ψ[#](...p|) with the modification based on γ and # M is the maximal sequence of signs: {1, 1, 1, ...} If we have two sign sequences A & B as well as two ordinals X & Y: X = Y if both are true: A = B Their components satisfy Xc = Yc X > Y either if: A ≥ B & Xc > Yc A > B & Xc = Yc For sequences A & B: A = B ⇌ A[η] = B[η] ∀η. A > B if there exists an ordinal η such that A[η] > B[η] —————— 3. Fundamental sequences: θ+1 = η & μ < cof(α) & δ in Suc & ρ in Suc ⋃ {0} α = Ω[η](ρ) & α[μ] = ψ(ρ, ψ(ρ+1, ..., ψ(ρ+μ, 0[η+μ]|), ..., [η+1]|)[η]|) & α[Ω[η](ρ)] = α & note: this also holds in the case of η in Lim as all ordinals like α are regular & 0 ≤ δ, λ, μ α = ψ(σ, β[η]|#) & # is non-empty: α = ψ(σ, β[η]|ρ,χ[φ],...,ρ(π),χ(π)[φ(π)]) & 0 ≤ π & (ρ,χ[φ]) → ψ(ρ,χ[φ]|) Single row demonstration: ψ(0[1]|β[0]) = ε(ψ(0[1]|)+1+β) & β < ζ(ψ(0[1]|)+1)*—Ω(ψ(0[1]|)+1)** Two row demonstration (Immediately generalizes): ψ(0[2]|5[1]β[0]) = ε(ψ(0[2]|5[1])+1+τ) & β < ζ(ψ(0[2]|5[1])+1)*—Ω(ψ(0[2]|5[1])+1)** —— α^^X = α^(α^^(X-1)) & α^^0 = 1 & α^^1 = α & X < ω κ is any structure composed of Ω[γ](δ) ordinals & ψ(Ω+2) → κ = Ω & δ = 2 Rules when p = 0 and F[θ](x) is the enumerated function based on η: β = σ = η = 0 & α[μ+1] = ω^α[μ] & α[0] = ω η > β = σ = 0 & α[μ+1] = F[θ](α[μ]) & α[0] = 0 β = δ & η = 0 & α[μ] = ω^(ψ(σ, δ-1[η]|)+1) & α[0] = 1 β = δ & η > 0 & α[μ] = F[θ](ψ(σ, δ-1[η]|)+1) & α[0] = 1 β = κ^^(n+1) & α[μ+1] = ψ(σ, (κ^^n)^α[μ][η]) & α[0] = 0 & n < ω β = κ^δ & α[μ+1] = ψ(σ, κ^(δ-1)*α[μ][η]|) & α[0] = 1 β = κ*δ & α[μ+1] = ψ(σ, κ*(δ-1)+α[μ][η]) & α[0] = 0 β = κ+δ & α[μ] = F[θ](ψ(σ, κ+δ-1[η]|)+1) β = Ω[γ](δ) & α[1+μ] = ψ(σ, G( ψ(δ, ..., ψ(δ+μ, Ω[γ](δ)[η]|), ..., [η]|))) & α[cof(α)] = ψ(σ,β) (10-12) γ ≠ η: G(x) = ψ(δ,x[γ]|) γ = η & δ ≠ σ: G(x) = x γ = η & δ = σ: α[μ+1] = ψ(σ, α[μ][η]|) & α[0] = ψ(σ,0) β = f(κ, Ω[γ](δ)) & α[μ+1] = ψ(σ, f(κ, ξ[μ])[η]) & ξ[μ+1] = ψ(δ, f(κ, ξ[μ])[γ]|) & ξ[0] = ψ(δ, 0[γ]|) β = λ & α[μ] = ψ(σ, λ[μ][η]|) (15, 16) cof(β) < Ω[η](σ): α[cof(β)] = ψ(σ, λ[η]|) cof(β) > Ω[η](σ): α[Ω[η]|(σ)] = ψ(σ, λ[η]) σ > β = 0 (18-23) σ in Suc & ξ+1 = σ & α[μ] = F[θ](Ω[η](ξ)+1)[μ] σ = Ω[η](0): α[μ+1] = ψ(α[μ], 0[η]|) & α[0] = ψ(0[η]|) σ < Ω[η](0) & σ in Lim & p = 0: α[μ] = ψ(σ[μ], 0[η]0|) σ < Ω[η](0) & σ in Lim & p = 1: α[μ] = F[θ](ψ(Ω[η](σ)+1)[μ] σ > Ω[η](0) & σ in Lim & σ ≠ Ω[#1](δ,p) & p = 0:  α[μ] = ψ(σ, 0[η]0|)[μ] using applied rules 1-17 on σ σ > Ω[η](0) & σ in Lim & σ = Ω[#](δ): ξ[μ+1] = ψ(ξ[μ], 0[η]|) & ξ[0] = ψ(δ[#]p|) & α[μ+1] = ψ(ψ(δ, ξ[μ]),0[η]|) & α[0] = ψ(0[η]|) —————— Examples of fundamental sequences: ψ[S](0) = ω ψ[S](α) = ω*(1+α) & α < ω*(1+α) ψ[S](ω^ω) = ψ[+](Ω) = ω^ω = Sup{1, ω, ω^2, ω^3, …} ψ[R](α) = Reg[α] & α < Reg[α] ψ[R](I) = ψ[R](M) = I(0) = I = Reg[1,0] (Inaccessibility) ψ[R](M+α) = Reg[1,α] & α < Reg[1,α] ψ[R](M+I(1)) = I(1) = ψ[Φ](M*2) = Reg[2,0] ψ[R](M*α+β) = Reg[α,β] & α < Reg[α,0] ψ[R](M*H) = ψ[R](M^2) = Reg[1,0,0] = H ψ[R]((M^(1+α))*(1+β)+γ) = Reg[α,β,γ] ψ[R](M^H(2)) = ψ[R](M^M) = Reg[1,0,0,0] ε(0) = ψ(0) = ψ[+](Ω^Ω) = Sup{ω, ω^ω, ω^ω^ω, …} ψ(α) = ε(α) & α < ε(α) ζ(0) = ψ(ζ(0)) = ψ(Ω) = Sup{ε(0),ψ(ε(0)),ψ(ψ(ε(0))), ...} min{ x | x = ψ(x)} ψ(Ω+1) = ε(ζ(0)+1) = Sup{ε(0), ψ(Ω)+1, ω^(ψ(Ω)+1), ω^ω^(ψ(Ω)+1), ...} min{ x | x = ψ(Ω)^x} ψ(Ω+2) = ε(ζ(0)+2) = Sup{ε(0), ψ(Ω+1), ω^(ψ(Ω+1)+1), ω^ω^(ψ(Ω+1)+1), ...} min{ x | x = ψ(Ω+1)^x} ψ(Ω+ζ(1)) = ψ(Ω*2) = ζ(1) = Sup{ε(0), ψ(Ω), ψ(Ω+ε(0)), ...} min{ x | x = ψ(Ω+x)} ψ(Ω^Γ(0)) = ψ(Ω^Ω) = Γ(0) = Feferman-Schütte Ordinal = Sup{ε(0), ψ(Ω^ε(0)), ψ(Ω^ψ(Ω^ε(0))), ...} min{ x | x = ψ(Ω^x)} ψ(ε(Ω+1)) = Bachman-Howard Ordinal = Sup{ε(0), ψ(Ω), ψ(Ω^Ω), ψ(Ω^Ω^Ω), ...} x > ψ(x) ψ(ζ(Ω+1)) = Sup{ε(0), ψ(ψ(1, 0)), ψ(ψ(1, ψ(1, 0))), ...} min{ x | x = ψ(ψ(1, x))} ψ(Ω(1)) = Sup{ε(0), ψ(ψ(1, 0)), ψ(ψ(1, ψ(2, 0))), ...} min{ x | x > ψ(ψ(1, x)) ψ(Ω(2)+Ω(1)) = Sup{ε(0), ψ(Ω(2)+ψ(1, 0)), ψ(Ω(2)+ψ(1, Ω(2)+Ω(1))), ...} min{ψ(x) > ψ(Ω(2)+ψ(1, x))} ψ(ψ(0|[1]|)) = Sup{ε(0), ψ(Ω), ψ(Ω(Ω)), ψ(Ω(Ω(Ω))), ...} min{ψ(x) = ψ(Ω(x))} ψ(ω, 0) = Sup{ε(0), ψ(1, 0), ψ(2, 0), ψ(3, 0), ...} ψ(ε(Ω(ω)+1)) = Takeuti-Feferman-Bucholz Ordinal = Sup{ψ(Ω(ω)), ψ(Ω(ω)^Ω(ω)), ψ(Ω(ω)^Ω(ω)^Ω(ω)), …} ψ(Ω, 0) = Sup{ε(0), ψ(ε(0), 0), ψ(ψ(ε(0), 0), 0), ...} min{ x | x = ψ(x, 0)} ψ(Ω(Ω)) = Sup{ε(0), ψ(Ω), ψ(Ω(ε(0))), ...} min{ x | x = ψ(Ω(x)))} ψ(Ω(1),0) = Sup{ψ(1), ψ(ψ(1), 0), ψ(ψ(1, ψ(1)), 0), 0), ...} ψ(Ω+1,0) = ε(Ω(Ω)+1) = Sup{Ω(Ω),Ω(Ω)^Ω(Ω),Ω(Ω)^Ω(Ω)^Ω(Ω), …}ψ(Ω*2,0) = {ψ(Ω,0), ψ(Ω+ψ(Ω,0),0), ψ(Ω+ψ(Ω+ψ(Ω,0),0),0), …} ψ(0[1]|) = Sup{Ω, Ω(Ω), Ω(Ω(Ω)), ...} min{ x | x = Ω(x)} ψ(1[1]|) = Sup{Ω(ψ(0[1]|)+1), Ω(Ω(ψ(0[1]|)+1), ..} min{ x | x = Ω(x) & x > ψ(0[1]|)} ψ(X+1[1]|) = Sup{ψ(X[1]|), Ω(ψ(X[1]|)+1), Ω(Ω(ψ(X[1]|)+1)), ...} min{ x | x = Ω(x) & x > ψ(Ω[1](0)[1]|)} ψ(Ω[1](0)^Ω[1](0)[1]|) = Sup{ψ(0[1]|), ψ(Ω[1](0)[1]|), ψ(Ω[1](0)^ψ(0[1]|)), ...} min{ x | x = ψ(Ω[1](0)^x[1]|)} ψ(0[1]3) = Sup{ψ(0[1]2), ω^(ψ(0[1]2)+1), ω^ω^(ψ(0[1]2)+1), ...} min{ x | x = ω^x & x > ψ(0[1]2)} or min{ x | x = ψ(0[1]2)^x} ψ(0[7]11[0]9[1]) = Sup{ψ(0[7]11[0]8[1]), Ω(ψ(0[7]11[0]8[1])+1), Ω(Ω(ψ(0[7]11[0]8[1])+1)), ...} min{ x | x = Ω(x) & x > ψ(0[7]11[0]8[1])} ψ(Ω(1)[1]|) = Sup{ψ(0[1]|), ψ(ψ(1, 0)[1]|), ψ(ψ(1, ψ(1, 0[1]|))[1]|), ψ(ψ(1, ψ(1, ψ(2, 0[1]|)[1]|))[1]|), ...} min{ψ(x[1]|) > ψ(ψ(1, x)[1]|)} ψ(Ω[1](1)[1]|) = Sup{ψ(0[1]|), ψ(ψ(1, 0[1]|)), ψ(ψ(1, ψ(2, 0[1]|)[1]|)[1]|), ...} min{ x | x > ψ(ψ(1, x[1]|)[1]|)} ψ(0[ω]|) = Sup{ε(0), ψ(0[1]|), ψ(0[2]|), ...} min{ x | x = Ω[λ](x) & λ < ω} ψ(0[Ω]|) = Sup{ε(0), ψ(0[ε(0)]|), ψ(0[ψ(ψ(0[ε(0)]|))]|), ψ(0[ψ(ψ(0[ψ(0[ε(0)]|)]|))]|), ...} min{ x | x = ψ(0[ψ(x)]|)} ψ(0[Ω+1]|) = Sup{ψ(0[Ω]|), Ω[Ω](ψ(0[Ω]|)+1), Ω[Ω](Ω[Ω](ψ(0[Ω]|)+1)), ...} min{ x | x = Ω[Ω](x)} ψ(0[Ω*2]|) = Sup{ψ(0[Ω]|), ψ(0[Ω+ψ(ψ(0[Ω]|))]|), ψ(0[Ω+ψ(ψ(0[Ω+ψ(ψ(0[Ω]|))]|))]|), ...} min{ x | x > ψ(0[Ω+ψ(x)]|)} —————— 4. Notes & Open Questions Let |x| be the cardinal associated to x: |ω| = ℵ(0) The following statements are taken to be true: Ω[1+α,0](κ,1)| = κ-th α-hyper-inaccessible cardinal Note that cof(Ω[ω](0)) = Ω[ω](0) > ω since the definitions for Ω[η]-indexed ordinals force them to be regular ∀η & it’s regularity only changes for δ-limit indices i.e cof(Ω[ε(0)](ω)) = ω but cof(Ω[ε(0)](4)) = Ω[ε(0)](4). We also take the following additional statements to be true: Stated in words: Consider adding a quantifier p that takes values {0,1}: if p = 0, treat it as a normal Ω[#]-indexed ordinal, regular or not, but if p = 1, do one of the following: If the index is a successor ordinal or zero, it is regular and equal to itself with p = 0 and therefore one can disregard p. However, if the index is a limit ordinal, move it into Reg and keep all its satisfied equations. Ω(6,1) = Ω(6,0) = Ω(6) & Ω(ω) = Ω(ω,0) < Ω(ω,1) Note that this is only modifies the ψ(α,...) function when α in Lim: A = ψ[#1](α,β[#2]0|#3) & B = ψ[#1](α,β[#2]1|#3) & C = ψ[#1](α+1,β[#2]|#3) A < B < C if α in Lim & A = B < C if α in Suc ⋃ {0} ψ[#1](α,β[#2]0|) = Enum{x | x = F(x)} < Ω[#1;#2](α,0) ψ[#1](α,β[#2]1|) = Enum{x | x = F(x) & x > Ω[#1;#2](α,0) if α in Lim} < Ω[#1;#2](α,1) with their enumerated function F(x) when α in Lim. Note that this forces ψ[#1](α,β[#2]1|) to include the ordinal Ω[#1;#2](α,0) & the function ψ[#1](α,β[#2]0|#3) in its construction set K & function set F respectively if α is in Lim. Example on ψ(α,β[0]p|)—Immediately generalizes: Ω(ω) = Ω(ω,0) = Sup{Ω, Ω(1), Ω(2), …} Ω(ω,1) = A = the ω-th regular cardinal = Reg[ω] ψ(ω,0) = B = Sup{ε(0), ψ(1, 0), ψ(2, 0), ...} < Ω(ω) If #2 = 0, then ψ[#1](β,σ[0]p|#3) → ψ[#1](β,σ\p|#3) ψ(0,β) = ε(β) & β < ζ(0)—Ω ψ(1,β) = ε(Ω+1+β) & β < ζ(Ω+1)—Ω(1) ψ(ω,β\0) = ε(B+1+β) & 0 < β < ζ(B+1)—Ω(ω) ψ(ω,β\1) = ε(Ω(ω)+1+β) & β < ζ(Ω(ω)+1)—A ψ(ω+1,β) = ε(A+1+β) & β < ζ(A+1)—Ω(ω+1) The following two statements should hold, assuming the truth of the previous statements. if α in Lim: ψ[#1](α,β[#2]0|#3) < Ω[#1;#2](α,0) < ψ[#1](α,β[#2]1|#3) < Ω[#1;#2](α,1) < ψ[#1](α+1,β[#2]1|#3) < Ω[#1;#2](α+1) if α in Suc ⋃ {0}: ψ[#1](α,β[#2]|#3) < Ω[#1;#2](α) < ψ[#1](α+1,β[#2]|#3) < Ω[#1;#2](α+1) ————— 5. Properties of the ψ function The ψ(β,α[#]p|) function collapses ordinals of higher cardinality to ordinals of cardinalities less than Ω[#](β,p) which are used for diagonalisation: ψ(α) = λ → {ω < λ < Ω} [Countable] ψ(α[#]|#2) = λ → {F[#] < λ < Ω[#](0)} ψ(β,α[#]|#2) = λ → {Ω[#](ξ) < λ < Ω[#](β) & ξ+1 = β} ψ(σ,α[#]0|#2) = λ → {Ω[#](ξ) < λ < Ω[#](σ) & ξ < σ} ψ(σ,α[#]1|#2) = λ → {Ω[#](σ) < λ < Ω[#](σ,1)} ψ(σ+1,α[#]|#2) = λ → {Ω[#](σ,1) < λ < Ω[#](σ+1)} All regular, and some singular, cardinals can also be enumerated in ψ-Form: ψ([α\0]1|) = α-th regular cardinal = Reg[α] ψ([α\1+β]1|) = α-th β-Inaccessible cardinal = I(β,α) & I(α) if β = 0 ψ([α,0@ξ\1+β]1|) = α-th β-hyper^ξ-inaccessible cardinal = I(β,0@ξ,α) & β > 0 To go access higher cardinals (Mahlo, Weakly Compact, etc.) one must consider the two functions Δ[Φ,L](α) and Φ(L,α) as well as the enumeration ordinal Ω[L](α). Δ(x) is any function from V[L], which is the set of all functions from ordinals to ordinals such that their output α makes Φ(L,α) = 0 for the statement L or more simply: their output doesn’t satisfy L. One can then define ψ[L](α) to be the enumerating function of all ordinals satisfying L. If α is an ordinal and L = A(0) & A(1) & … & A(n)—that is, multiple sub-statements, then satisfying L means Φ(A(m),α) = 1 ∀m. Multivariate ordinal functions are handled slightly differently in V[L]. A 2-variable function f(α, β) in is V[L] for at least one fixed variable α or β, if there is sum remainder function g(δ) that would is completely in V[L]. Ω[L](σ) and ψ[L](σ, α) are defined as follows: ψ[L](σ, α) = min{ z | z not in K(σ,α) over F} K[0](σ,α) = {α, σ} ⋃ {Ω[L](σ)} K[n+1](σ,α) = K[n](σ,α) ⋃ {f(x) | f in F & x in K[n](σ,α)} K(σ,α) = ⋃[q in N]{K[q](σ,α)} F = {ψ[L](η, ξ), Δ[Φ,cof(δ) < δ](β)| β,ξ,η in K[n](σ, α) & ξ < α & σ ≤ η & Δ in V[L]} Ω[L,P](σ) = Enum[min]{ x | x > ψ(σ, y) & Φ(L,y) = Φ(P,x) = 1 & Φ(P,ψ(η, y)) = 0 & η ≤ σ ∀y in Ord} V[L] = { f: Ord → Ord | f(x) = y & Φ(L,x) in {0,1} & Φ(L,y) = Φ(P,y) = 0 ∀y in R(f) & R(g) = the range of g} If a statement involves an ordinal function, then it is also included in V[L] with the restriction that their output must doesn’t satisfy L. Examples: L = β in Lim → ψ[S]-function & Ω is uncountable V[L] = {α+1} Ordinal multiplication isn’t in V[L] since there is no remainder function that would be in V[L] for any fixed value even if a non-collapsing restricted version is used. L = β = ω^β → ψ[Ord]-function & Ω is regular V[L] = {α+ξ → ξ > 0, α*ξ → 1 < ξ ≤ α & ω^α}
 * all arguments ≥ 0 unless otherwise stated*
 * 1) represents anything that is within delimited brackets ∀η > 0
 * min{ x | x = ε(x) & x > ψ(0[1])}
 * min{ x | x = ε(x) & x in Reg & x > ψ(0[1])}
 * min{ x | x = ε(x) & x > ψ(0[2]5[1])}
 * min{ x | x = ε(x) & x in Reg & x > ψ(0[2]|5[1])}
 * Ω[1+α](κ,1)| = κ-th α-inaccessible cardinal with η = 0 it collapses and equals the (1+κ)-th regular cardinal since the first is ℵ(0).
 * Ω[1+α,0@β](κ,1)| = κ-th α-hyper^β-inaccessible cardinal, with β hyper’s.
 * ω| = ℵ(0) & Ω[#](α) = ℵ[#](1+α)—α in Suc ⋃ {0}, or the Generalized Continuum Hypothesis is true if and only if α in Suc ⋃ {0} that is: 2^ℵ(α) = ℵ(α+1) when α in Suc ⋃ {0}.
 * Ω[#](α,0)| = ℵ[#](α) < |Ω[#](α,1)| = ℵ[#](α)* < |Ω[#](α+1)| = ℵ[#](α+1)—α in Lim, or the Generalized Continuum Hypothesis is false when α in Lim.
 * ω| = ℵ(0) = First infinite regular cardinal = Reg[0]
 * Ω(0)| = ℵ(1) = Second infinite regular cardinal = Reg[1]
 * Ω(ω,0)| = ℵ(ω) & cof(Ω(ω,0)) = cof(ℵ(ω)) = ω
 * Ω(ω,1)| = ω-th regular cardinal = Reg[ω]
 * Ω(α,1)| = Reg[1+α] ∀α > 0
 * Ω(ω+1)| = (ω+1)-th regular cardinal = Reg[ω+1]
 * Ω[1](0)| = I(0) = First inaccessible cardinal
 * Ω[1](ω,1)| = I(ω) = ω-th inaccessible cardinal
 * Ω[1](ω,0)| = a large fixed point of x = ℵ(x)
 * Ω[1](ω+1)| = I(ω+1) = (ω+1)-th (0-)inaccessible cardinal
 * Ω[7,0](0)| = First 6-hyper-inaccessible cardinal
 * Ω[1,0,0](ω+6)| = (ω+6)-th hyper^2-inaccessible cardinal
 * Ω[1;0](κ,1)| = κ-th regular cardinal satisfying x = hyper^x-inaccessible