Eventually I'll make a large number using these definitions, but I know that I need to ensure that there's nothing wrong with them first, so I would appreciate if you would tell me anything wrong with these definitions:
I define a system as a collection of axioms paired with a collection of statements and what they evaluate to.
A system has a constructability of 0 iff it is turing complete and no more powerful than a turing machine. A system has a constructability of n (n>0,n∈⍵) iff the highest constructability system it can simulate and get the largest number definable in any x∈⍵ amount of symbols has an n-1 constructability.
For example FOST if paired with a set of axioms such as ZFC would be a valid system. I think that it wouldn't have a constructability though, since it does have the ability to simulate the BB function, but I'm pretty sure it can't specify axioms.