User blog comment:Bubby3/Guesses about the strength of BMS and the Catching Hierarchy./@comment-32697988-20180419222547/@comment-30754445-20180420091713

Well, any ordinal can be translated into a hydra, if you make the rules complex enough.

The same is true for arrays. Array notations aren't designed to reach the first stable ordinal (which is - indeed - synonymous to Π_n reflection), yet strong array notation seems to get there. How? Because it has a very very very complex set of rules.

BMS, on the other hand, is known for the simplicity of its rule set. It isn't surprising that it reaches  BHO at (0,0)(1,1) and ψ(Ωω) at (0,0,0)(1,1,1), because those are natural ordinals to reach for hydras.

And I haven't really seen any evidence that BMS gets anywhere near stable ordinals. I'm not talking about proof, but any kind of evidence''. Stable ordinals are huge''. I doubt there are more than three or four people here who have an actual understanding of them (and I'm certainly not one of them).

Why is this relevant? Because people tend to grossly underestimate ordinals they don't understand. This is true at all levels. When people look "beyond", they tend to intuitively extrapolate from concepts that they know well, and when it comes to ordinals this is a very bad idea.

A person who is only familiar with knuth up-arrows, for example, cannot begin to imagine how Veblen functions work. His mind is not yet ready for anything beyond simple diagonalization. His vague mental image of (say) Γ₀ would probably be closer to ω2.

Similarly, a person who is only familiar with two-variable Veblen functions, cannot begin to imagine how OCF's work. A person who is only familiar with Madore psi, cannot begin to imagine how inaccessible cardinals work.

And the jumps from I to M to K to Pi-4 reflection to stables are much more difficult. I only understand an inkling of processes involved, but that's more than enough for me to be at awe of just how big these ordinals are. There are no shortcuts here. And unless a person knows exactly what they're doing, they're practically guaranteed to mess things up at these levels (by which I mean: they'll grossly underestimate everything).

BTW I've recently tried to work out for myself how Deedlit's Mahlo notation works. It's downright incredible. An entire new world. People should really regard these huge ordinals with more awe and respect. If you think it is easy, then you're doing it wrong.