well, hopefully, somebody will get the kinks out and make this possible someday the net is still (relatively) new when you think about it.
I appreciate there is no such thing as infiniteth. Still the numbers get longer, so the memory required to hold them would get longer. Until eventually you ran out of electrons in the the universe with which to address it. Also while your set S maybe infinite, again putting S into a RAM module isnt practically useful. While that set could describe such a system you could individually identify a computer with it.
That is why the internet is not infinite. If it was, you would run out of memory, even though each of the infinite number of IP address has an address of finite length.
I like the internet, but I do think that it could be a potential danger. Of course, it could help us start a revolution as well. Anyway, I'm interested to see what the internet will be like in 30 years...