Paul Tanner III >The term can be used in more than one way. When I said "name" I meant, for sake of simplicity as I explicitly said, instead of saying (x,0) just say x.
You don;t understand. There aren't enough name likes x, x, z, x', x'' etc. Those are countable -- leading to a terribly unfair game of musical chairs. The vast majority of reals are unnameable and undefinable, lost souls waiting for someone kindhearted to pity them.
Your "process" is no process at all, it has 0% chance of being carried out in general. It has a big "gotcha!" that you blithely skip over ("A Miracle Happens Here!)
Even phrases like "W.L.O.G." won't make this pig sing.
For the countable set of numbers that are amenable to a process leading to an answer (find the product of two numbers,) you will find iteration or recursion absolutely necessary one way or another.
In defense of similar triangles I will say I'm not against imaginary procedures that cannot really be carried out. They may even lead someone to carry out a simulacrum of the unreal procedure in the real world. The real world versions only work for a finite number of limited precision numbers though, and in the engraving of various scales (that word!) an iteration procedure was involved somewhere along the line.