I have followed the argument that rationals, being countable and ordered, can be covered by a convergent sequence of decreasing intervals.
I am trying to understand why the same argument can’t be applied to the reals, on the basis that axiom of choice leads to the well ordering theorem which says that the reals can be well ordered. In that case why can’t I cover the first real with an interval length ½ etc in the same way as the rationals ?
I suspect that there aren’t enough natural numbers to complete the process since the reals are uncountable, but can the process not extend into the ordinal ω1. If it does then surely the series ½ + ¼ ... should still sum to 1 ? I’d appreciate any feedback or web reference you could offer to clarify this for me.
The same argument can't be applied to the reals since a sum over uncountably many positive numbers will always be infinite. Thus, regardless of the size of these "open intervals" under the well ordering, there are necessarily uncountably many with a positive radius. Therefore, the sum of their measures will be infinite.
Edit: At @user43208's suggestion, the details to prove the above statement goes as follows: if $S$ is a set of positive reals and the supremum over all sums of finite subsets is finite, then each set $S_n=\{x\in S:x>1/n\}$ is finite, hence $S$ (the union of the $S_n$) is countable.