[FOM] Ordinary Mathematics: CHH x ZF?

[Zuhair Abdul Ghafoor Al-Johar]

What parts of Ordinary mathematics are formalizable in ZF that are not formalizable in the Cumulative Hereditary Hierarchy "CHH" defined below?

H_0=0
H_i+1={x| x h_=< H_i}  where i+1 is an ordinal that is a member of H_i for all i>2
H_j=UH_i for all i<j when j is a limit ordinal that is a member of some H_i

where "h_=<" stands for hereditarily subnumerous.

 CHH is strong enough up to the first cardinal fixed point. So Borel determinacy is formalizable in it. 

Zuhair