User blog comment:Hyp cos/Attempt of OCF up to Stability/@comment-35470197-20181026043805/@comment-35470197-20181027113756

> Does this method work?

It works if you first give a system of fundamental sequence which is not necessarily monotonic, but does not give an ordinal notation system.

> Or just simply take the expressions in generation of \(C_i(\alpha,\beta)\), and the "<" is the same as the "<" of ordinals themselves?

The binary relation "<" is uncomputable unless you give a premitive recursive replacement, as many authors of papers on OCFs do. Maybe you skipped to read this, which I mentioned above.

So your OCFs do not generate computable large numbers.