This area is still confusing. I think actual definitions would help more than a few examples, especially if you're deviating from the standard Veblen function.

@SimplyBeautifulArt made another attempt at cleaning up the notation. i would add definitions, but i'm not really sure how (i'm such a noob #facepalm)

@meowzz Have you read the above conversations? If nothing else, I at least think my descriptions of the Veblen function are intuitive and more precise than giving a few examples. It's not clear still what φ''(0) is, and definitely not clear to me what φ''(1) is.

"If you want to convey your ideas to others, you should learn to precisely express them." is me right now

@SimplyBeautifulArt do you have any references or futher explanation of "Namely, instead of just a map from indices to parameters you have a map from indices to maps!"?

"here's a completely differently structured kind of ordinal collapsing function that behaves more like the Veblen function " which?

my thinking / understanding is as follows: in the og veblen there is no way to write the LVO (please correct me if im wrong). so in my notation, that would be the point you switch to φ''(0)

@meowzz That was the extended Veblen function that I had made on a whim.

@meowzz still doesn't explain what φ'' is in general

@meowzz afraid you probably wouldn't want to look into it for now

Extending the Veblen function the way user21820 and I were describing is fairly straightforward if you understand the way I was defining the Veblen function.