User blog comment:Allam948736/Ordinal hyperoperators and BEAF - analysis/@comment-35470197-20200120225654/@comment-35470197-20200121083150

I searched on twitter, and then I found informations on more extensions. I do not know the explicitly definition for extensions 2～5. (I have just heard evaluations or as rough explanations as the blog post above.) I have heard that several on them are in discord.
 * 1) ω↑↑(ω+1) = ε_0.
 * 2) ω↑↑(ω+1) = ε_0×ω.
 * 3) ω↑↑(ω+1) = ε_0^ω.
 * 4) ω↑↑(ω+1) = ε_1.
 * 5) ω↑↑(ω+1) = φ(ω,0).

Unless we explicitly define α↑^{γ}β by transfinite induction, we can set ω↑↑(ω+1) in any desired way, as the OP did. Therefore explaining "how transfinite hyperoperator should work" without an explicit definition is usually meaningless.