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Nept and Nopt structures (Part 2).
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Nept and Nopt structures (Part 2).
It can be noticed that the induced SeedValues from a NOPT Structure have a similar role to the omega (w) limit which is used as a “default way” of describing the length of an infinite list. I think that omega is a shorthand for saying “we don’t know how long an unending list of items goes on for, but we’ll call it omega”. This seems more sensible than saying “yes we do know, the list goes on for as many natural numbers there are”. This is because, while the process of generating naturals (into Standard Positional Notation) is well-defined and clear, the extent of the natural numbers depends on how creative we get by combining powerful operations with NOPT structures. In other words, the set of Natural Numbers is well-defined at the level of (+1)-iterative description, but not at the level of unbounded descriptive capabilities.
As the examples shown above demonstrate, there are plenty of intermediary ellipsis stages that large numbers need to pass through in order to reach an elusive target magnitude goal.
The omega limit is used for limit ordinals in theory of infinite ordinals, but it is independent of the values of the symbols in the list.
So lim<w^1, w^2, …> = w^w, and is understood as an w-limit.
And lim<w, w^w, w^w^w, …> = e0, and is understood as an w-limit.
And lim<e0, e0^e0, e0^e0^e0, …> = e1, and is understood as an w-limit.
And lim<e0, e1, e2, …> = e_w, and is understood as an w-limit.
In the last example, it doesn’t matter that all the component symbols are much larger than omega - we have “large machinery at our disposal” (eg e0) – but we always use the “small infinite value” of omega to decide the list length of any of these infinite sequences whenever a limit to the sequence is desired. Omega is like a convenient yardstick that is used in evaluating limit points for horizonal phenomena.
Whenever a list structure emerges and a limit is desired then the idea that an infinite list of symbols can be indexed or sequenced by the natural numbers is called upon.
So in theory of infinite ordinals, omega is the convenient yardstick.
In NOPT structures, the induced SeedValues have the “omega role” by deciding how long a stage should be lingered upon before the transitioning to other levels becomes necessary.
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