if there existed two 0s 01=x, 02=0 then 01+02=01. then define now 02=x, 01=0, 02+01=02. then Apostol seems to take these 2 equations and aruge that by commutativity 01+02=02+01 and therefore there is a unique 0? but I don't see how this proves the hypothesis by contradiction. haven't you proven only that x=x by tautology with 01+02=01, 03+02=03, 01=03? maybe I'm missing something in the proof.

I guess a universe with 2 0s would be kind of interesting like polarized to positive and negative 0