Could someone please point out errors in this reasoning?
I am aware and understand the points of both Cantor's diagonal argument and theorem, but:
Suppose we take an incremental list of sets of natural numbers, such as {1}, {1, 2}, {1, 2, 3}, ... etc.
and then we form sets of subsets of each consecutive steps. In principle, we could account for all of the subsets that could be formed in the course of exhausting N in this way, although the number of those subsets does rise exponentially.
On the other hand, if we make a list of consecutively longer decimals such as 0.x, 0.xx, 0.xxx..., etc., in which 'x' is a placeholder for numbers, and we make a new list containing all of combinations of numbers 0-9 out of each of those elements on the original list, we would, if the lists continue to be filled up forever, account for all of the reals between 0 and 1.
Infinities in both of these cases are only potential, and it is assumed that the construction process could go on forever for it to yield the desired result.
What is wrong with this? Thanks!
I'll focus first on the second argument. Your construction using decimals of the form 0.x, 0.xx, 0.xxx, etc. will exhaust all real numbers with finitely many decimal places, even if that finite number is arbitrary. This is a subset of the rational numbers. You need to consider reals with infinite precision. To give a concrete example, your construction will never output $\pi$.
With regards to subsets of the natural numbers, a similar thing is happening. Your algorithm will only ever output a finite subset of the natural numbers, and will never, for example, output the set of even natural numbers. In order for your procedure to work, it must output every possible subset of $\mathbb{N}$ at some point. Thus, you have to tweak your procedure to allow for the creation of infinite subsets. Cantor proved, however, that you will never be able to make this work.