User blog comment:Rgetar/Three properties of standard forms/@comment-35470197-20181124090716/@comment-35470197-20181124223835

> "compute x[n]": here x is already given in some form, and we already have algorithm of calculating fs element. So, we get only one fs of x.

I see. So you do not have to mind the problem on the choice of a fundamental sequence. However, the ill-definedness derived from the fact that there is no algorithm to compute \(y[n]\) from \(y = f(x)\) (without using \(x\)) if \(f\) is not injective is still valid here.

In order to avoid this problem, you need to define a fundamental sequence without using \(f\), or introduce an ordinal notation system associated to your OCFs so that you can directly deal with expressions.