User blog comment:Edwin Shade/The Grand List Of Transfinite Ordinals/@comment-30754445-20171130012046

"every significant ordinal"?

Sorry, but that's a logical impossibility. There are an infinite number of them. Worse: there are an infinite number of them below ω₁.

I also think that "eventually superseding anything anyone has ever devised before" is not a realistic goal. Professional mathematicans have been at this for decades, and niether you or me or anybody else here is going to beat them in their own game.

(They may not be interested in googology par se, but ordinals have other more "legitimate" uses)

But don't let this discourage you! There's great value in trying to get as high as you can (and as a side note: it is more important to understand each step along the way, than it is to plunge forward to great heights we may not be ready for)