
8
This blog post will present three guesses about collapsing cardinals beyond weakly compacts: one for admissibles using Scorcher's results, one for admissibles using my results, and one for uncountables using Scorcher's results. We will compare with BMS.
Let =~ mean Ï‰)stable ordinal
 [WIP]

Time to get started.
In Z_{2}, we can't talk about transfinite ordinals, but we can talk about wellorderings of natural numbers. To prove that an ordering of order type x is wellordered, we have to prove the following: take some formula Ï†(n) such that Ï†(0) and if Ï†(x) for all x < y in the ordering, Ï†(y); then, prove that âˆ€n(âˆ€mâ‰¤n in the ordering(Ï†(m))).
For a wellordering of order type Ï‰^Ï‰, we are going to use this encoding:
 0 corresponds to 0.
 1 corresponds to 1.
 P[a]P[b]P[c]...P[z] corresponds to 1+Ï‰^{a1}Ï‰^{b1}Ï‰^{c1}...Ï‰^{z1}.
[WIP]
Update: So I've heard about these things called which I think is a system that assigns a tree to every formula transformed using rules in a similar format of those of CoC. For some consistent theory X, the trees that can bâ€¦
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I am making a new blog post series in which I will analyze progressively stronger theories and try to find lower bounds for their strength. The goal of this series is to try to unify the higher portion of the two branches of computable googology, so we can know if, say, BMS is stronger than Friedman's functions derived from finite promise games.
Here are the theories that I am planning on analyzing:
 Z_{2}
 Z_{3} (I need help with the definition of this one)
 Higher Order Arithmetic (I need help with the definition of this one)
 ZFC
 ZFC + there exists an inaccessible cardinal
 ZFC + some extensions to inaccessible cardinals
 ZFC + there exists a Mahlo cardinal
 ...
 ZFC + there exists a rankintorank cardinal
After that, I will analyze the previous systems with prâ€¦
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Oh look, it's another one of those guides for beginner googologists! But this one isn't a long series of blog posts or a website or whatever. This one just explains the basic ruleset of Googology: if you understand and apply these ru, you can learn the rest (the major googological notations, methods to create googological notations, the major googologism families...) by searching for them on this wiki and studying them.
So, without further ado, here are the rules:
 Don't try to create the largest number ever. Yet. Seriously, don't. Unless you're familiarized with set theories and how to extend them, you should NOT try to create the largest number yet.
 Don't use Infinity anywhere in your number/notation (a function that doesn't need to be made â€¦

To avoid confusion with my other lambda notation, this one uses a lowercase psi instead of an uppercase psi.
Define the set F of seminormal functions using these inductive rules:
 F(x) = 0 Ïµ F
 F(x) = x Ïµ F
 If F() Ïµ F, then G(x) = S(F(x)) Ïµ F (where S is the successor function)
 If F(), G() Ïµ F, then H(x) = F(x)+G(x) Ïµ F
 If F(), G() Ïµ F, then H(x) = F(x)G(x) Ïµ F
 If F(), G() Ïµ F, then H(x) = F(x)^{G(x)} Ïµ F
 If F(), G() Ïµ F, then H(x) = Ïˆ_{Î»F(x)}(G(x)) Ïµ F
 If F() Ïµ F, then G(x) = Î»_{F(x)}{n} Ïµ F for all nonnegative integers n
Define the set of normal functions as F âˆ© F(x) = 0. Then, define xâ†‘â†‘1 = x, xâ†‘â†‘(n+1) = x^{xâ†‘â†‘n}, and (xâ†‘â†‘Ï‰)[n] = xâ†‘â†‘(Ï‰[n]). Then, define climb(n) = n, but all Î»_{a}{b}s inside of n have been changed to Î»_{a}{b+1}s, and finally, define fs(F(Î»_{x}{n}),0)â€¦
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