I am no formal logician but I guess your argument implies that one assumes Zermelo-Frankel's set theory (otherwise one can devise a finite collection of "sets"

I am no formal logician but I guess your argument implies that one assumes Zermelo-Frankel's set theory (otherwise one can devise a finite collection of "sets" and "functions" for which your "axiom" does not hold). Now, if you are in ZF, then you have infinite sets and as far as I know, your argument is true only if you accept the axiom of choice (which you may or may not but is not a "trivial" thing). Also notice that the mathematical world is not exactly the same as the "real" world... Let us imagine there are parts of the Universe which are farther than any possible interaction with them. Are they reduced to a single point? Why not? Would really like to receive an answer.