User blog comment:DrCeasium/Continued hyperfactorial array notation/@comment-5150073-20130327205450/@comment-5150073-20130329104224

To define the recursive notation that goes up to Church-Kleene ordinal, we need an infinite number of types of symbols. Types, not symbols. This is because we can have sequence of symbols S_1, S_2, S_3, ... but we can have the symbol B_1 that diagonalized over all. Otherwise, that notation will be not recursive.