User blog comment:P進大好きbot/Summary on historical background of BMS/@comment-30869823-20180713171021/@comment-35470197-20180803220709

> Number theorists most definitely did ignore Fermat's conjecture... with the sole exception of looking for ways to prove it (or at least, prove it for a given subset of exponents).

I could not understand what you mean. Number theorists did not ignore the conjecture, which that was not a theorem at all at the time.

> We should put our efforts into actually showing that it works (or showing that it doesn't work). More to the point:

Exactly.

> How about this for a first baby step: > Let's show that BM1 indeed reaches Γ₀ at (0,0)(1,1)(2,1)(3,1).

The baby step is to show that the primitive sequence reaches \(\varepsilon_0\). It is very very easy for googologists working on BMS, but is there a formal proof of it? One of the reasons why there is no proof of the statement that BMS reaches \(\textrm{BHO}\) is because most of them do not know how to write a formal proof. I guess that even if it is pretty easy, they have no idea to prove the strength of the primitive sequence. (Of course, I guess that you can do. I am talking about googologists who do not know what a formal proof means.)

> (since I'm actually well-versed in BM1, I might try my hand at doing this)

Great. I am also trying to verify the strength of the pair sequence system for BM1.1, BM2, and BM2.3 with respect to Kotaitan's classification.

...Well, I started trying 11 days ago. But I found errors in his decriptions of the classifications many times. Finally, I guess that there is no error in his description yesterday.

At least, I have finished to write down a (personal, rough, but completely formalisable sketch of a) proof for the babiest step, i.e. the primitive sequence system. Since I am not so good at analysing as googologists here, I will spend more days to work on the pair sequence system.

I think that it is good for you to show such a baby trial for other googologists in order to tell them how to write a formal proof. ...I should? I am afraid that even if I do, few googologists "believe" my formal proof because I am a newbie here. Or I am afraid that they say "It is completely trivial. We don't know why you need a formal proof for it. Or are you a really baby?", because my activiy has essentially no effect here. Instead, you, who is well-known to be a professional googologist here, have sufficient influences.

> With BMS, as you stated yourself, the system itself isn't well defined. So the first step, really, should be to agree on a specific set of expansion rules for each version. A definition by a computer program is fine... as long as the source code is openly available, and as long as everybody agrees which precise program defines which version (and as long as the program actually compiles).

Exactly. Even if I complete formal proofs for Koteitan's classification, they would say that "do the definitions really define the expansion rules of BMS?" That is the worst problem on this topic for me. The "actual" behaviour of BMS might depend on each googologist, as long as we refer to a version which is not written in formulae or programming.

> But with the exception of BM1, we don't have even that. So yes, Fejfo is quite right that without an agreed-upon definition, the whole thing should be ignored.

The "should be" has no legal forces against freedom of googology. That is why I said that it depends on each person.

> (Do you think anybody would have given Fermat's Last Theorem any attention, if they had to actually guess what Fermat meant?)

Hmm...? I (and perhaps also almost all other mathematicians) do not have much interest in Fermat himself. We are just respecting the powerful formula \(\forall n \geq 3, \neg(\exists (x,y,z) \neq (0,0,0), x^n+y^n=z^n)\), its proofs, and its appliciations. I do not know about Fermat.