User blog comment:Scorcher007/SLCON and almost 3000 comparisons, largest comparison table with DAN/@comment-33713741-20181218173744/@comment-31580368-20181219021138

What does HYPERCOUNTABLE mean? Can you give a definition?

Ordinal #3 can already be considered HYPERCOUNTABLE, because it is ∆11 subsets (hyperarithmetic subsets) of ω.

Or maybe Ordinal #2778? Axiom about the existence of this ordinal in Kripke–Platek set theory makes the theory as strong as the axiom about the existence of 1st uncountable ordinal. That is, this ordinal seems to be countable, but if we replaced it with an uncountable one, then for the theory nothing would have changed. Therefore, it is called smallest admissible which is  not  locally  countable. The same applies to the ordinal #2790 just for ZFC without powerset axiom.

You can still talk about {zoo 2.25}? Ordinal that is ∆12 subsets of ω and the least ordinal with such property is the ζ in ITTM (the supremum of all eventually writable ordinals). According to Madore this ordinal is bigger that model of ZFC. And it looks like it's incredibly big. My notation cannot describe this ordinal. I suppose he should be between #2824(ordinal that model of ZFC without powerset plus n-reflective cardinal) and #2824 (ordinal that model of ZFC without powerset plus measurable cardinal)