I will extend the Denis Maksudov's extended Buchholz's function beyond the omega fixed point, to beat the limit of the extended Buchholz's function.

Unfortunately, this extension has been abandoned with the statement: "It's very hard to define the extension".

Introduction

/// WIP ///

DeepLineMadom introduces my simplification of ψ0 as simply ψ, and extends the ordinal collapsing function beyond the Extended Buchholz's ordinal (ψ0(Ω_Ω_Ω_...)).

Inaccessible cardinal

We introduced the inaccessible cardinal, I, which diagonalizes over Ω_α and Rathjen's Phi function, where Φ(1,0) denotes the least omega fixed point, hence:

• ψI(0) = Ω

• ψI(1) = Ω_2

• ψI(2) = Ω_3

• ψI(3) = Ω_4

• ψI(ω) = Ω_ω

• ψI(ω+1) = Ω_(ω+1)

• ψI(ψI(0)) = Ω_(ψI(0)) =  Ω_Ω

• ψ(I) = ψ(ψI(ψI((...ψI(0)...)))) = ψ(Φ(1,0))

• ψ(I+1) = ψ(Φ(1,0))*ω

• ψ(I+Ω) = ε(ψ(Φ(1,0))+1)

• ψ(I+Ω_2) = ε(ψ(Φ(1,0))+Ω_2)

• ψ(I+ψI(I)) = ψ(Φ(1,0)2)

• ψ(I+ψI(I)2) = ψ(Φ(1,0)3)

• ψ(I+ψI(I)ω) = ψ(Φ(1,0)ω)

• ψ(I+ψI(I)^2) = ψ(Φ(1,0)^2)

• ψ(I+ψI(I)^ω) = ψ(Φ(1,0)^ω)

• ψ(I+ψI(I)^ψI(I)) = ψ(Φ(1,0)^Φ(1,0))

• ψ(I+ψI(I+1)) = ψ(I+Ω(ψI(I)+1)) = ψ(ε(Φ(1,0)+1))

• ψ(I+ψI(I+2)) = ψ(I+Ω(ψI(I)+1)) = ψ(ε(Ω(Φ(1,0)+1)+1))

• ψ(I+ψI(I+ω)) = ψ(I+Ω(Φ(1,0)+ω))

• ψ(I+ψI(I+ψI(I))) = ψ(I+Ω(Φ(1,0)2))

• ψ(I+ψI(I+ψI(I+1))) = ψ(I+Ω(ε(Φ(1,0)2+1)))

• ψ(I+ψI(I+ψI(I+ψI(I)))) = ψ(I+Ω(Ω(Φ(1,0)+1)))

• ψ(I2) = ψ(Φ(1,1))

• ψ(I2+ψI(I2)) = ψ(Φ(1,1)2)

• ψ(I2+ψI(I2+ψI(I2))) = ψ(Ω(Φ(1,1)2))

• ψ(I3) = ψ(Φ(1,2))

• ψ(I4) = ψ(Φ(1,3))

• ψ(Iω) = ψ(Φ(1,ω))

• ψ(IΩ) = ψ(Φ(1,Ω))

• ψ(IψI(I)) = ψ(Φ(1,Φ(1,0)))

• ψ(IψI(IψI(I))) = ψ(Φ(1,Φ(1,Φ(1,0))))

• ψ(I^2) = ψ(Φ(2,0))

• ψ(I^2+I) = ψ(Φ(1,Φ(2,0)+1))

• ψ(I^2*2) = ψ(Φ(2,1))

• ψ(I^3) = ψ(Φ(3,0))

• ψ(I^4) = ψ(Φ(4,0))

• ψ(I^ω) = ψ(Φ(ω,0))

• ψ(I^Ω) = ψ(Φ(Ω,0))

• ψ(I^ψI(I)) = ψ(Φ(Φ(1,0),0))

And finally, we reached:

• ψ(I^I) = ψ(Φ(1,0,0)), which diagonalizes over all levels of binary Rathjen's Phi function level.

Moving on:

• ψ(I^I+I) = ψ(Φ(1,Φ(1,0,0)+1))

• ψ(I^I+I^2) = ψ(Φ(2,Φ(1,0,0)+1))

• ψ(I^I+I^ψI(I)) = ψ(Φ(Φ(1,0,0),1))

• ψ(I^I+I^(ψI(I)+1)) = ψ(Φ(Φ(1,0,0)+1,0))

• ψ(I^I*2) = ψ(Φ(1,0,1))

• ψ(I^I*ω) = ψ(Φ(1,0,ω))

• ψ(I^I*ψI(I^I)) = ψ(Φ(1,0,Φ(1,0,0)))

• ψ(I^(I+1)) = ψ(Φ(1,1,0))

• ψ(I^(I+1)*2) = ψ(Φ(1,1,1))

• ψ(I^(I+2)) = ψ(Φ(1,2,0))

• ψ(I^(I+ω)) = ψ(Φ(1,ω,0))

• ψ(I^(I+ψI(I^I))) = ψ(Φ(1,Φ(1,0,0),0))

• ψ(I^(I2)) = ψ(Φ(2,0,0))

• ψ(I^(I3)) = ψ(Φ(3,0,0))

• ψ(I^(Iω)) = ψ(Φ(ω,0,0))

• ψ(I^(I*ψI(I^I))) = ψ(Φ(Φ(1,0,0),0,0))

• ψ(I^I^2) = ψ(Φ(1,0,0,0))

• ψ(I^I^2*2) = ψ(Φ(1,0,0,1))

• ψ(I^(I^2+1)) = ψ(Φ(1,0,1,0))

• ψ(I^(I^2+I)) = ψ(Φ(1,1,0,0))

• ψ(I^(I^2*2)) = ψ(Φ(2,0,0,0))

• ψ(I^I^3) = ψ(Φ(1,0,0,0,0))

• ψ(I^I^4) = ψ(Φ(1,0,0,0,0,0))

• ψ(I^I^5) = ψ(Φ(1,0,0,0,0,0,0))

The limit of the finitary Rathjen's Phi function is ψ(I^I^ω). At this point we need to define the "successor cardinal" of I. That's easy - it's Ω(I+1). Then we can have further successor and, eventually limit cardinals. Here are some examples of how these are used:

• ψ(Ω(I+1)) = ψ(ε(I+1))

• ψ(Ω(I+1)2) = ψ(ε(I+2))

• ψ(Ω(I+1)I) = ψ(ε(I2))

• ψ(Ω(I+1)^2) = ψ(ζ(I+1))

• ψ(Ω(I+1)^Ω(I+1)) = ψ(Γ(I+1))

• ψ(Ω(I+2)) = ψ(ε(Ω(I+1)))

• ψ(Ω(I+3)) = ψ(ε(Ω(I+2)))

At this point we reach another sequence where most major ordinals are the same, like ψ(Ω(I2)) and ψ(Ω(Ω(I+1))).

Now we reach the higher inaccessible cardinals which diagonalizes over all previous inaccessible cardinals, like ψ_I2(0) = I. The I_α-fixed point is I(1,0). It can be done similarly to Veblen's phi function and Rathjen's Phi function.

To continue briefly, see https://googology.fandom.com/wiki/List_of_systems_of_fundamental_sequences.