Turing's topological proof that every written alphabet is finite (2010) 2 years ago
I have no idea about topology, so sorry if my question is stupid: Assume we are limited to the unit square (0<=x<=1 and 0<=y<=1). Now I map every real number in [0,1] to a symbol by simply defining the symbol for the number r as x=r and y=r. This results in a uncountable infinite amount of symbols. Now it's clearly not true that it's possible to describe any of those new symbols with a finite amount of symbols from a finite alphabet (because otherwise this would also be possible for real numbers). Where is my intuition wrong?