First, one may ask "What is the strongest function in googology, if we allow unformalised stuffs?" There are many candidates such as "Sam's function", "Croutonillion function", or something like that, but the answer unfortunately heavily depends on the interpretations of those works. Instead, I would like to consider a more specific question:
Which is regarded as a stronger notation: BEAF or UNOCF?
The reason why I chose two is because they have many common features:
- Both of them are very famous and popular.
- Both of them are known to be unformalised.
- Both of them have many intuitive analyses by experts.
- Both of them had so many adherents, and still have a few adherents.
- Both of them were considered to be the strongest computable googolism.
- Both of them were said to be easy to formalise, although nobody has ever succeeded in the formalisation.
- Both of the limits of their agreed-upon expansions correspond to \(\varepsilon_0\).
- Both of their expansions above \(\varepsilon_0\) depend on their adherents.
- Both of their standard forms above \(\varepsilon_0\) depend on their adherents.
In this blog post, I roughly collect "estimations" of UNOCF and BEAF in this wiki. The winner will be the one which has a greater estimation. Thank you for spending time to read my April fool blog post.
UNOCF[]
According to a testimony, UNOCF surpasses DAN. Since DAN is also considered to be very strong if we ignore the problem that it is also unformalised, it ensures how strong UNOCF is considered to be.
In addition, UNOCF is considered to surpass all OCFs. Therefore it surpasses Buchholz's OCF, Extended Buchholz's OCF, Rathjen's OCF with M, Rathjen's OCF with K, Stegert's OCF, Arai's OCF, Bashicu's OCFs, and so on. Since one of Bashicu's OCF is intended to surpass \((0,0,0,0)(1,1,1,1)\) in BM2.3, it ensures how powerful UNOCF is considered to be.
Moreover, the stage cardinal \(T\) in UNOCF is considered to play a role of a very strong large cardinal such as the least \(\Pi^2_0\)-indescribable, and extra symbols such as \(X\) and \(H\) allow UNOCF to go even further. For example, my own extension of UNOCF is intended to surpass \(\textrm{PTO}(\textrm{ZFC})\).
BEAF[]
On the other hand, according to another testimony, BEAF surpasses Rathjen's OCF with M or K at \(\{L,X,1,X \uparrow \uparrow X\}n,n\), and surpasses an OCF based on stability at \(\{L2,1\}n,n\).
Oh, but UNOCF even goes further, as I explained above. So, is BEAF smaller than UNOCF?
Please wait. According to another testimony, BEAF surpasses any computable googolism. Since UNOCF is considered to be a computable googolism, it directly implies that BEAF surpasses UNOCF!
- Winner: BEAF
I note that BEAF even surpasses BEAF, because BEAF itself is considered to be a computable googolism.