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Abbreviation: KSS (Kinshasa sequence system; formerly Kinshasa array notation (KSAN) / Kinshasa sequence notation (KSSN), now the part of the sequence system)
Created in July 2022. This sequence system is defined to create transfinite ordinals using brackets, addition, bracket natural number multiplication, bracket nesting, and transfinite recursion.
The sequence system is named after the city of Kinshasa, Democratic Republic of the Congo (DR Congo).
The structure of the sequence is based on the brackets, ironically referred to the megacity of Kinshasa with more than 10 million people.
It comes in at least 2 parts:
Primitive sequence system (PSS) - Introduces sequence brackets and summation of ordinals. FGH level ε0 with respect to the Wainer hierarchy.
Pair sequence system (2SS) - Introduces 2-entry sequence brackets. Intended FGH level ψ0(Φ_1(0)) with respect to the extended Buchholz's function, where Φ_1(0) indicates the least omega fixed point.
Trio sequence system (3SS) - Introduces 3-entry sequence brackets.
Finitary sequence system (FSS) - Allows 4 or more entries in the sequence brackets.
Extended finitary sequence system (XFSS) - Introduces more expansion rules into the respecting separators.
Using this sequence system, there are many ways to create a computable large number using varities of the notations.
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.
The Kinshasa sequence notations are currently composed of 1 notation, which is as follows (let @ to be any Kinshasa sequences):
Kinshasa hyperoperator notation (KHN) [Argument type: binary; format: a{@}b]
I plan to create the following notations in the future:
Kinshasa power sequence notation (KPSN) [Argument type: unary; format: @[a]]