Take cantors set of points on a line [0,1] if we have a function that maps the points onto a list we will always find a number not on that list through the diagonal argument. Yet if we keep taking the midpoints of [0,1] to a list wont we eventually fill every point. it would look like this:
- 0.875 and so on...
I understand that its impossible for fractions to represent an irrational number yet with each repetition you are forming a new series that will converge towards every irrational number between [0,1]. Surely as the list tends to infinity i could argue that the fractions will converge to the irrational numbers? i apologize i don't have the mathematical rigor to formally construct the argument.