User blog comment:Cloudy176/Hyperfactorial array notation: Analysis part 1/@comment-1605058-20130507200045/@comment-5150073-20130508195552

If Hollom's notation is ill-defined, then how Cloudy has been able to compare it with FGH ordinals (which are well-defined)?

Anyways, after a few months of studying recursion, I concluded that any well-defined recursive notation must have at least 3 general rules. First is "iterating and preceeding rule" (example is the catastrophic rule in BEAF), second is "chopping rule" (that chops default levels of recursion), third is "terminating rule" (eventually all expressions will come to this rule). They can be dropped down to smaller rules, since they can be thought as concepts rather than true rules. For Hollom's notation we have the first such concept (in 2 rules):

n![k,@] = ((...(n![k-1,@])![k-1,@]...)![k-1,@])![k-1,@], with n '![k-1,@]'s

n![1,...,1,1,k,@] = n![1,...,1,[1,...,1,1,1,@],k-1,@]

Both conditions are exist: we have the iteration and preceeding (k is decreased by 1).

Chopping concept is actually this one (in two rules):

n![1] = n!n

n![@,1] = n![@]

Terminating concept in his notation can be found in somewhat strange way: it depends on up-arrow notation. Nonetheless, the following rule can be thought as terminating:

n!m = n{m}(n-1){m}(n-2){m}...{m}4{m}3{m}2

{m} is a shorthand for ^^^...^^^ (m up-arrows).

Thus we can conclude that, using this informal way, Hollom's hyperfactorial array notation is well-defined.