Good morning Dejan!
You are right and I agree: Every real differs from all possibe FISs - _but not by nodes or digits_. Every real number needs a finite definition! (And there are only countably many.)
The FISs or digits are only tools to express a real number as well as possible. In case that only zeros follow after a certain digit, this expression is even exact, like 0.25 = 1/4. In case that only 3's follow, we can escape to the ternary system and have there 0.1 = 1/3. In case of irrational numbers we have no chance to express them exactly. We can only use as many FISs as we like. Exac…
You are right and I agree: Every real differs from all possibe FISs - _but not by nodes or digits_. Every real number needs a finite definition! (And there are only countably many.)
The FISs or digits are only tools to express a real number as well as possible. In case that only zeros follow after a certain digit, this expression is even exact, like 0.25 = 1/4. In case that only 3's follow, we can escape to the ternary system and have there 0.1 = 1/3. In case of irrational numbers we have no chance to express them exactly. We can only use as many FISs as we like. Exac…