The primitive sequence system is the first part of the Kinshasa sequence system.
The sequence system consists of brackets with one argument, 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" 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:
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) = () = 1, n = (0)(0)(0)...(0)(0)(0) with n copies of (0)'s
Tailing rule: x+0 = x
Nesting rule: (x)[b] = (x[b])
Multiplication rule: x0 = 0, x1 = x, xn = x+x+x+...+x+x+x with n copies of x's
Recursion rule: (x+1)[b] = (x)b = (x)+(x)+(x)+...+(x)+(x)+(x) with n copies of (x)'s
On the other hand, multiplication rule is valid if the right-hand side of the sequence is a natural number (e.g. ((1)3) is not equal to ((1)+3), but ((1)(0)) is equal to ((1)+(0))).
Moreover, concatenated brackets containing multipliers such as (2)2(1) is also equal to (2)(2)(1) too!
If there are two or more distinct rules applicable to one expression, the lowest-numbered rule which is applicable will be applied.
Let the special function E is defined as:
E[0] = 0
E[n+1] = (E[n])
We can see that E[1] = 1, and E[n] = ((((...((((0))))...)))) with n pairs of brackets.
Since the sequence system itself is uncomputable, we need to create the sequence notation (KS[]) associated to the fundamental sequences of the respective system, i.e. a recursive interpretation of the comparison and the system of fundamental sequences of the sequence system using formal expressions, in order to create a computable large number, as seen as the definition above.
Formally speaking, the normal form of the sequence system is only created with zero (0), bracket concatenation (as an ordinal addition), and bracket nesting (as transfinite recursion of bracket concatenation). Zeroes can be omitted. Say, the normal form of 0 is 0, the normal form of 1 is (0) or just (), the normal form of (1) is ((0)) or (())
Here is the rule of the sequence system based on normal form (modified from the former one):
Base rule: 0[b] = 0, (0) = 1, a[b] = a where "a" is a sum of (0)'s with "a" copies of (0)'s, x+a[b] = x+a where "a" is copies of (0)'s
Tailing rule: x0 = x
Nesting rule: (x)[b] = (x[b])
Recursion rule: (x+(0))[b] = (x)b = (x)+(x)+(x)+...+(x)+(x)+(x) with n copies of (x)'s
We can see that the normal form of natural numbers are n = (0)(0)(0)...(0)(0)(0) with n copies of (0)'s, and the normal form of multiplied brackets are (x)n = (x)(x)(x)...(x)(x)(x) with n copies of (x)'s.
6 = (0)(0)(0)(0)(0)(0)
(4) = ((0)(0)(0)(0))
((2)3) = (((0)(0))((0)(0))((0)(0)))
((((1)))) = (((((0)))))
The primitive sequence system uses pretty much similar to the Cantor normal form.
Comparison for transfinite ordinals with Cantor normal form:
((0)) = (1) = ω
(1)+1 = ω+1
(1)+2 = ω+2
(1)+3 = ω+3
(1)+(1) = (1)(1) = (1)2 = ω2
(1)2+1 = ω2+1
(1)2+2 = ω2+2
(1)3 = ω3
(1)4 = ω4
((0)(0)) = (2) = ω^2
(2)+1 = ω^2+1
(2)+2 = ω^2+2
(2)(1) = ω^2+ω
(2)(1)2 = ω^2+ω2
(2)2 = ω^2*2
(2)3 = ω^2*3
(3) = ω^3
(4) = ω^4
(((0))) = ((1)) = ω^ω
((1))+1 = ω^ω+1
((1))(1) = ω^ω+ω
((1))(2) = ω^ω+ω^2
((1))2 = ω^ω*2
((1))3 = ω^ω*3
((1)+1) = ω^(ω+1)
((1)+2) = ω^(ω+2)
((1)2) = ω^ω2
((1)3) = ω^ω3
((2)) = ω^ω^2
((2)2) = ω^(ω^2*2)
((3)) = ω^ω^3
((4)) = ω^ω^4
(((1))) = ω^ω^ω
(((1)2)) = ω^ω^ω2
(((2))) = ω^ω^ω^2
(((3))) = ω^ω^ω^3
((((1)))) = ω^ω^ω^ω
((((2)))) = ω^ω^ω^ω^2
(((((1))))) = ω^ω^ω^ω^ω
((((((1)))))) = ω^ω^ω^ω^ω^ω
The limit of the primitive sequence system is ε0 with respect to the Cantor normal form (Wainer hierarchy if using notations or functions).
(3)[4] = (2)4 = (2)+(2)+(2)+(2) = (2)(2)(2)(2)
((1)(1))[5] = ((1)(0)5) = ((1)+5)
((((1))))[3] = ((((0)3))) = (((3)))
E[6] = (E[5]) = ((E[4])) = (((E[3]))) = ((((E[2])))) = (((((E[1]))))) = ((((((E[0])))))) = ((((((0)))))) = (((((1)))))