User blog comment:Pellucidar12/Attempt at a FGH related notation/@comment-29915175-20170416151157/@comment-30754445-20170418181627

I'm aware of Kleene's O notation.

The problem is that such nonalgorithmic definitions have very little practical value. For one thing, we don't know how to write small ordinals in such a system. For another thing, the FS won't make much intuitive sense. For example, we have no reason to believe that ω would expand to 1,2,3,4...

I suppose one could argue that we don't have a formal proof that all nonalgorithmic definitions will have these drawbacks. But given that we have no idea how to calculate even BB(6), I think it is safe to say that the odds of achieving thing goal without some huge conceptual breakthrough in mathematics are close to nil.