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I'm really really sorry, but this is incomplete. I have been making a text document (up to 42kb rn) trying to formalise and explain a Googological System, but it was never finished. I've not edited it in two months, and thought it was a better idea to let it die here than on my hard drive.
Still, I'm pretty sure there are some interesting and salvageable ideas here. Since writing it, I've learned a lot of type theory (in particular, I think I have a decent understanding of inductiverecursive types), so if I come back here at some point, I should be able to clean up the presentation and present what I have in an inductive, not recursive, manner. (I'd probably also switch from math notation to computing notation, too. Easier to work with in â€¦
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Terms are identified up to equivalence â‰¡. A model is consistent iff K â‰¡ SK is false.
 Sabc â‰¡ ac(bc)
 Kab â‰¡ a
 If ac â‰¡ bc, c is a free variable and c does not occur in a or b, a â‰¡ b
The extensions additionally include some of the following terms. It is not guaranteed that all extensions or combinations of extensions are consistent.
 Qab â‰¡ K if aâ‰¡b and Qab â‰¡ SK otherwise
 Eab â‰¡ K if there is some c such that ac â‰¡ b, and Eab â‰¡ SK otherwise
 Cab â‰¡ Kc for some c such that ac â‰¡ b (if it exists), and Cab â‰¡ SKK otherwise
 Mabc â‰¡ K if there is some d such that bd â‰¡ c and ad â‰¡ K, and Mabc â‰¡ SK otherwise
 Nabc â‰¡ Kd for some d such that bd â‰¡ c and ad â‰¡ K, and Nabc â‰¡ SKK otherwise
First off, K and SK represent the truth values true and false, respectively. Given this, it is somewhat clear what each ofâ€¦
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Literally. As of writing, there are 10,787 pages on this wikia, and 11,084 articles in the numbers category.
I'm mainly using this as an excuse to pick up the discussion which was dropped about a month ago from the comments here. Please have a quick read of them if you haven't yet.
I believe the two popular ideas were to:
 A) Consolidate similarsized numbers from the same notation into one article each, possibly with a section on each number
 B) Create a splinter site or forum, probably nonwikia based.
I want to get this discussion going again. Both ideas appeal to me and honestly, I want to see us getting together and doing something towards these possible goals. It's better than deciding stuff should happen but not following through, so hopefâ€¦
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1)
\(B_0(\alpha,\beta) = \beta \cup \{0,1,\omega,\Omega,\Omega^+\}\)
\(B_{n+1}(\alpha,\beta) = \{\gamma
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The ish is because I've never seen something like this presented, except for some type theories, which are much less useful.
Anyway.
Primitive recursion is a simple method for creating total recursive functions. We can formalise it in such a way that (almost) any sufficiently slowgrowing (sub \(\omega\) on the FGH) googological function can be encoded, and hence offers a way of proving that it is recursive and total.
This can even be represented in a way that is easily understandable: zero, successor, projection and composition can be paid no heed when showing a function is PR, and only the recursion step needs to be stated explicitly  and even this can be represented in a simpler manner.
Similarly, a simple extension of primitive recursionâ€¦
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