User blog comment:Kyodaisuu/English description of Fish numbers/@comment-10429372-20131102144937/@comment-1605058-20131102165730

Ikosarakt, yes, this is exactly as you say, BB hierarchy can be easily extended. We can use "diagonalized" oracle which will work as BB_w, and so on up to w_1

Kyodaisuu, Taranovsky's notation is very hard to comprehend, many mathematicians are trying to understand it and check if it is as strong as author thinks.