User blog comment:Ubersketch/Hypernomial Hierarchy v3/@comment-35470197-20190703221954

Ill-defined.

What does \(f(n)\) mean? Is it a function with a variable symbol \(n\)? What is the domain? A formal strings? The result of a function with input \(n\)? Then what is \(n\)? What is \(a[n]\)? Is it a formal strings? Or do you apply an operation \([n]\) in a recursive way?

You need to clarify the meanings of symbols using quantifiers. For example, write something like "For any map \(f \colon \mathbb{N} \to \mathbb{N}\), \(f \in h\)". Otherwise, other can read "For any \(f \colon \Omega \to \Omega\)", "For any function symbol \(f\) in a fixed language", and so on.

Also, the comparison with \(\omega\) is undefined.

You might be thinking that writing mathmetics-like sentences is sufficient, but it is wrong. If you are so sophisticated in mathematical formula that you can clearly declare all quantifiers in formulae, it is actually sufficient. But you know that you are not, because you are always pointed out error. Therefore it is reasonable to write down detailed explanations about what symbols mean.

Please learn the writing methods from other googologists' blog post. Please do not skip this elementary step. Honestly, it is better to understand that your recuent stuffs are always ill-defined, and hence you need to learn more from others.