User blog:PsiCubed2/An example that shows why a "table of ordinals" proves nothing

Consider the BMS expression:

(0,0,0)(1,1,1)(2,1,0)(1,1,1)

How would we analyze this?

A "quick and dirty" approach will give us:

(0,0,0)(1,1,1) = ψ(Ω_ω)

(0,0,0)(1,1,1)(1,1,1) = ψ(Ω_ω * 2)

(0,0,0)(1,1,1)(2,0,0) = ψ(Ω_ω * ω)

(0,0,0)(1,1,1)(2,1,0) = ψ(Ω_ω * Ω)

(0,0,0)(1,1,1)(2,1,0)(1,1,1) = ψ(Ω_ω * Ω + Ω_ω)

On the other hand, if you create a more detailed table (as found here, for example), you will get ψ(Ω_ω^2). I actually checked Aarex's table carefully, and every step on the way seems to makes sense.

But the "quick and dirty" version also makes sense. Having a table that makes sense, apparently, is no guarantee that the table is correct.

So why should we believe that the more detailed tables are right? How can be we be sure that an even more detailed analysis won't give us yet another answer? After all, even Aarex's table skips lots and lots of ordinals.

That's the problem with creating tables without any explanation or reasoning. The problem is that when we're dealing with such large ordinals, it is simply impossible to cover all the important cases. And using "intuition" to fill in the gaps is a very risky business.