User blog comment:Denis Maksudov/Slowly growing ordinal function and FS up to BHO./@comment-24920136-20170404223258

I see you are trying to simplify FS by rewriting OCF such that one can write all countable ordinals such that a<BHO in the form Psi(beta) where Beta is written in NF with base Ω

While developing Egg Notation i have came across a similar problem when developing a future extension to Egg Notation, the approach i've taken is to have a hybrid FS system that uses both phi and psi ( or NF base w + NF base Ω)

1: Have the closure operations include the veblen function, then psi(0) = the limit of the veblen function to the amount of variables we allow, for example, if we let phi(alpha,beta) be in there, then psi(0) = Gamma_0, this is by definition. So we can define FS for all delta < Gamma_0 and then have FS based on psi for ordinals higher. Since the "limit" of one function is the 0 start of the next, there will be no overlap

2.- If we chose not to have phi as a closure function but instead a -> w^a, in this case we must simply include FS for CNF base w for ordinals below psi(0)/epsilon_0, then psi(alpha) for the larger ordinals.

Note that with NF Ω, only addends and exponents must be allowed to also be base Ω, for this reason, coefficients can be expected to be countable and thus writeable using the phi/psi hybrid, and this forms the framing for a larger system if you want to extend the whole thing further