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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
by k92mueller 2y 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?
- ainoobler 2y agoHow can you tell the difference between (r,r) and (r+ε,r+ε)? The argument in the article assumes that there is an ε such that it is impossible to tell the difference between r and r+ε. So this means that the entire square can be covered by squares of side length ε and since the unit square is compact this means that there are only finitely many ε squares required to cover the entire unit square. Variation within the ε squares is imperceptible so this means there are only finitely many symbols that can be perceived to be different.
- constantcrying 2y ago>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. Every symbol has measure zero and so every symbol is identical.