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A successor of Kinshasa sequence system to primitive sequence system.

Definition

The sequence system consists of brackets with two arguments, whole numbers (zero (0) and natural numbers (1, 2, 3, ...), i.e. 0, 1, 2, 3, ...), plus signs (+), brackets addition and nesting, and sequence multiplication using natural numbers. Zeroes (0) can be omitted and plus signs can be omitted only if one of the sequence is followed by an another sequence.

Say, bracket concatenation is the sequence addition, and bracket nesting is the sequence transfinite recursion.

Let "x", "y", and "z" represent any sequence of the primitive sequence.

Let the format of the system is a[b], where "a" be a sequence, and "b" be a whole number (non-negative integer) which runs through fundamental sequences. Here are the rules:

1. Base rule: 0[b] = 0, a[b] = a where "a" is a natural number, x+a[b] = x+a where "a" is a natural number, (0,0) = (0) = () = 1, n = (0,0)(0,0)(0,0)...(0,0)(0,0)(0,0) with n copies of (0,0)'s

2. Shorthand rule: (x,0) = (x)

3. Tailing rule: x+0 = x

4. Nesting rule: (x)[b] = (x[b]) if x < ((0,1),0)

5. Multiplication rule: x0 = 0, x1 = x, xn = x+x+x+...+x+x+x with n copies of x's

6. Finite recursion rule: (x+1)[b] = (x)b = (x)+(x)+(x)+...+(x)+(x)+(x) with n copies of (x)'s

7. Uncountable recursion rule:

   1. z = 0: ((x,y+1),0)[0] = 0, ((x,y+1),0)[b+1] = (((x,y+1),y)[b],0)

   2. z > 0: ((x,y+1),z)[0] = ((0,y),z), ((x,y+1),z)[b+1] = (((x,y+1),y)[b],z)

Similar to the primitive sequence system, two transfinite sequence brackets can be written either in two ways, with or without plus signs in bracket addition.

If there are two or more distinct rules applicable to one expression, the lowest-numbered rule which is applicable will be applied.

Fundamental sequences

The fundamental sequences of the PSS and 2SS are different. You can see that the fundamental sequence of (((0,1),1),0)[n] is not (((0,1),1),0)[0] = 1, (((0,1),1),0)[n+1] = ((((0,1),0)[n],1),0), but (((((0,1),1),0)[n],1),0).

Here are fundamental sequences of each milestone sequences (the OCF comparison is extended Buchholz's function!):

• ((0,1),0)

  • Fundamental sequence: ((0,1),0)[0] = 0, ((0,1),0)[n+1] = (((0,1),0)[n],0)

  • Buchholz's function: ψ0(ψ1(0)) = ψ0(Ω)

  • Veblen function: φ(1,0) = ε0

• ((0,1)(0,1),0) = ((0,1)2,0)

  • Fundamental sequence: ((0,1)(0,1),0)[0] = 0, ((0,1)(0,1),0)[n+1] = ((0,1)((0,1)(0,1),0)[n],0)

  • Buchholz's function: ψ0(ψ1(0)+ψ1(0)) = ψ0(Ω2)

  • Veblen function: φ(1,1) = ε1

• ((1,1),0)

  • Fundamental sequence: ((1,1),0)[0] = 0, ((1,1),0)[n] = ((1,1)[n],0) = ((0,1)n,0)

  • Buchholz's function: ψ0(ψ1(ψ0(0))) = ψ0(Ωω)

  • Veblen function: φ(1,ω) = εω

• (((0,1),1),0)

  • Fundamental sequence: (((0,1),1),0)[0] = 0, (((0,1),1),0)[n+1] = (((((0,1),1),0)[n],1),0)

  • Buchholz's function: ψ0(ψ1(ψ1(0))) = ψ0(Ω^2)

  • Veblen function: φ(2,0) = ζ0

  • A.k.a. Cantor's ordinal

• (((0,1)(0,1),1),0) = (((0,1)2,1),0)

  • Fundamental sequence: (((0,1)(0,1),1),0)[0] = 0, (((0,1)(0,1),1),0)[n+1] = (((0,1),1)(((0,1)(0,1),1),0)[n],0)

  • Buchholz's function: ψ0(ψ1(ψ1(0)+ψ1(0))) = ψ0(Ω^3)

  • Veblen function: φ(3,0) = η0

• (((1,1),1),0)

  • Fundamental sequence: (((1,1),1),0)[0] = 0, (((1,1),1),0)[n] = (((0,1)n,1),0)

  • Buchholz's function: ψ0(ψ1(ψ1(ψ0(0)))) = ψ0(Ω^ω)

  • Veblen function: φ(ω,0)

• ((((0,1),1),1),0)

  • Fundamental sequence: ((((0,1),1),1),0)[0] = 0, ((((0,1),1),1),0)[n+1] = (((((((0,1),1),1),0)[n],1),1),0)

  • Buchholz's function: ψ0(ψ1(ψ1(ψ1(0)))) = ψ0(Ω^Ω)

  • Veblen function: φ(1,0,0) = Γ0

  • A.k.a. Feferman-Schütte ordinal

• ((((0,1)(0,1),1),1),0)

  • Fundamental sequence: ((((0,1)(0,1),1),1),0)[0] = 0, ((((0,1)(0,1),1),1),0)[n+1] = ((((0,1)((((0,1)(0,1),1),1),0)[n],1),1),0)

  • Buchholz's function: ψ0(ψ1(ψ1(ψ1(0)+ψ1(0)))) = ψ0(Ω^Ω^2)

  • Veblen function: φ(1,0,0,0) = Γ0

  • A.k.a. Ackermann ordinal

• ((((1,1),1),1),0)

  • Fundamental sequence: ((((1,1),1),1),0)[0] = 1, ((((1,1),1),1),0)[n] = ((((0,1)n,1),1),0)

  • Buchholz's function: ψ0(ψ1(ψ1(ψ1(ψ0(0))))) = ψ0(Ω^Ω^ω)

  • A.k.a. small Veblen ordinal

• (((((0,1),1),1),1),0)

  • Fundamental sequence: (((((0,1),1),1),1),0)[0] = 1, (((((0,1),1),1),1),0)[n+1] = (((((((((0,1),1),1),1),0)[n],1),1),1),0)

  • Buchholz's function: ψ0(ψ1(ψ1(ψ1(ψ1(0))))) = ψ0(Ω^Ω^Ω)

  • A.k.a. large Veblen ordinal

• ((0,2),0)

  • Definition: ((0,2),0) = (((0,2),1),0)

  • Fundamental sequence: ((0,2),1)[1] = (0,1), ((0,2),1)[n+1] = (((0,2),1)[n],1) => ((0,2),0)[1] = ((0,1),0), ((0,2),0)[n+1] = (((0,2),1)[n],0)

  • Buchholz's function: ψ0(ψ2(0)) = ψ0(Ω_2)

  • A.k.a. Bachmann-Howard ordinal

• (((0,2),2),0)

  • Fundamental sequence: (((0,2),2),1)[1] = (0,1), (((0,2),2),1)[n+1] = (((((0,2),2),1)[n],2),1) => (((0,2),2),0)[1] = (((0,1),2),0), (((0,2),2),0)[n+1] = (((((0,2),2),1)[n],2),0)

  • Buchholz's function: ψ0(ψ2(ψ2(0))) = ψ0(Ω_2^2)

• ((0,3),0)

  • Fundamental sequence: ((0,3),2)[1] = (0,2), ((0,3),2)[n+1] = (((0,3),2)[n],2) => ((0,2),0)[1] = ((0,2),0), ((0,3),0)[n+1] = (((0,3),2)[n],0)

  • Buchholz's function: ψ0(ψ3(0)) = ψ0(Ω_3)

• ((0,(1,0)),0)

  • Fundamental sequence: ((0,(1,0)),0)[0] = ((0,0),0), ((0,(1,0)),0)[n] = ((0,(0,0)n),0) = ((0,n),0)

  • Buchholz's function: ψ0(ψ_{ψ0(ψ0(0))}(0)) = ψ0(ψ_ω(0)) = ψ0(Ω_ω)

  • A.k.a. Buchholz's ordinal

• ((0,(1,0)(0,0)),0) = ((0,(1,0)+1),0)

  • Fundamental sequence: ((0,(1,0)(0,0)),(1,0))[0] = (0,(1,0)), ((0,(1,0)(0,0)),(1,0))[n+1] = (((0,(1,0)(0,0)),(1,0))[n],(1,0))

  • Buchholz's function: ψ0(ψ_{ψ0(ψ0(0))+ψ0(0)}(0)) = ψ0(ψ_{ω+1}(0)) = ψ0(Ω_(ω+1))

  • A.k.a. Takeuti-Feferman-Buchholz ordinal

• ((0,(0,1)),0)

  • Fundamental sequence: (0,(0,1))[0] = 1, (0,(0,1))[n+1] = (0,((0,(0,1))[n],0))

  • Buchholz's function: ψ0(ψ_{ψ_{ψ0(0)}(0)}(0)) = ψ0(Ω_Ω)

• ((0,(0,(0,1))),0)

  • Fundamental sequence: (0,(0,(0,1)))[0] = 1, (0,(0,(0,1)))[n+1] = (0,(0,((0,(0,(0,1)))[n],0)))

  • Buchholz's function: ψ0(ψ_{ψ_{ψ_{ψ0(0)}(0)}(0)}(0)) = ψ0(Ω_Ω_Ω)

And finally, the countable limit of the pair sequence system is ψ0(Ω_Ω_Ω_...) = ψ0(ψ_{ψ_{ψ_{...}(0)}(0)}(0)) = ψ0(Λ) where Λ denotes the least omega fixed point. The limit is known as the countable limit of extended Buchholz's function!!! YIKES!!!