User blog comment:Ikosarakt1/Coding strings as ordinals (for Friedman's n(k))/@comment-5029411-20130729191800/@comment-1605058-20130729201756

Let's add one symbol - 0. We have 2 types of well formed sentences: 0 and XYZ, where X is a letter and Y and Z are well formed. Let AYZ mean phi(0,Y)+Z, BYZ mean phi(1,Y)+Z, CYZ mean phi(2,Y)+Z, etc. Completeness proof - each ordinal below phi(w,0) has Veblen normal form psi(a,b)+psi(c,d)+... where a,c are finite and b,d are in normal form. By transfinite induction if we have b and psi(c,d)+... defined by formulas Y and Z, respectively, then XYZ denotes our ordinal, where X is a+1-th letter. Nonuniqueness proof - A00 denotes psi(0,0)=1, AA000 denotes psi(0,1)=w, A0AA000 denotes psi(0,0)+psi(0,1)=1+w=w, so w has at least 2 representations.