I extended 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

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.