First, what is an "usual" number-naming scheme? We take a certain hierarchy, , and map a word for it up to certain n and (not necessarily do it for each ). The restriction is only that the rule always keeps. So, under that, Bowers' scheme is usual:
= biggol (even though we have no word for , it still works)
Let's define so-called mixed-arrow notation as follows:
- is an expression composed by 's and 's.
Rule 1. (# is empty, no arrows at all)
Rule 2. (b=1, regardless of #)
Rule 3. (# ends at up-arrow)
Rule 4. (# ends at down-arrow)
Note that because they both drop to multiplication which is associative.
Unusual naming scheme
Now we want to form names for this unique notation.
Let be 0 and be 1. Then we can map a binary expansion of each number to the arrow-expression. For example, 5 = 1012 = . So we can form the prefix for our names, adopting English numbers. For the suffix, I propose "-arrowal". For a and b we choose "3" as the smallest non-trivial case.
So it forms the names for the numbers as follows:
- Zerarrowal = = 27
- Onarrowal = = 27 (not greater than previous term)
- Twarrowal = = 19683
- Thrarrowal = = 7625597484987
- Fourarrowal =
- Fivarrowal =
- Sixarrowal =
- Sevarrowal = (exactly tritri in Bowers' system)
- Eigarrowal =
To estimate eigarrowal we have to determine how fast grows. Consider it:
. So eigarrowal is about , which is, as we know, much less than sevarrowal = . It turned out that in our naming scheme (n+1)-arrowal can be even lesser than n-arrowal.
- Ninarrowal =
- Tenarrowal =
- Thousarrowal (superscripts still can be used).
I see no way to pin down this sequence to a standard ordinal hierarchy .