User blog comment:Alemagno12/BM2 has a problem/@comment-30754445-20180724073311/@comment-35470197-20180725094348

@PsiCubed2

How about introducing another terminology meaning "analysis with proofs"? I guess that "analysis" in recent googology here just means "analysis with sufficiently many lines in tables carefully written by reliable googologists" or something like that. Therefore even if you state that the "analysis" is informal, few googologists might understand what the problem is.

Personally, I also wish that every analysis were given in a formal way, because it is not fair. Even if I defined an ordinal notation beyond BMS with formal proof, I could never go beyond ones without proofs or precise definitions, because it is impossible to compare objects with formal arguments and objects with no precise arguments. Any "desire" can be true in an informal way. Any "dream" admits implicit axioms beyond \(\textrm{ZFC}\) such as \(0=1\). For example, if one uses a large cardinal and state "it can be replaced by a recursive analogue under \(\textrm{ZFC}\)!" without proofs, none might go beyond his/her large number under \(\textrm{ZFC}\). Such informal arguments are as strong as uncomputable large numbers.

It looks similar to the difference between computable large numbers and uncomputable large numbers. None requires Rayo's function to be computable, and googologists mainly study computable functions even though they are obviously tiny compared to other uncomputable large functions. Googologists understand the existence of two or more regulations in googology.

Namely, there are two googologies on computable large numbers. One is a googology with formal proofs. The other one is a googology with reliable tables written by reliable googologists. None requires formal proofs here even though their arguments possibly lack the accuracy.

Also, there are merits and demerits in the acceptance of informal arguments.

Merits: If googologists allow any informal arguments, then they can study googology, as high school students study real numbers without the definition of \(\mathbb{R}\). They feel satisfied. Beginners feel relaxed to understand how to write "analysis". These merits are very important for them, and should not be prevented, I think.

Demerits: It lacks the accuracy. Some feel unsatisfied. Beginners are robbed of an opportunity to learn the significance of formal proofs. It rejects the possibility for googologists with formal proofs to go beyond ones without formal proofs. (It might be a merit for the majority, though. It helps "top googologists" to stay top.)

@Rpakr

If you have a proof but do not know the way to write it in a formal way, then I can translate your proof in a completely formal way. I believe you!

Anyway, writing a sufficiently big table is just an evidence, but not a formal proof.

For example...
 * Riemannian hypothesis has a table with \(10^{13}\) lines of evidence, but has not verified yet. If one wants to show Riemannian hypothesis, then he/she needs to show that "every" non-trivial zero is on the critical line.
 * Collatz conjecture has a table with \(10^{18}\) lines of evidence, but has not verified yet. If one wants to show Collatz conjecture, then he/she need to show that "every" Collatz sequence terminates. Even if Collatz sequences gives a monstrous ordinal notation beyond BMS, no one can prove it now.
 * Easier example. If you want to prove that the image of Euler's function \(x^2 + x + 41\) is contained in the set of prime numbers, then you need to show that "every" value is a prime number. Even if you write a table with dozens of lines, it does not give a formal proof.