If you allow yourself the infinite set of the natural numbers, you can pretty easily get an infinite set of sets. For instance, for every natural N, take the set of naturals less than N. Or greater than N, if you'd like an infinite set of infinite sets.
As the article notes, these particular infinite families are easy to produce a choice function for: just take the least element of each set. But not all infinite families have enough determined structure for that kind of rule.
If we tried to make it impossible to construct an infinite family of sets, we'd have to disallow relatively reasonable families like the ones I described above. Those are pretty useful, though, so it makes sense to address the problem further downstream.
(I suppose another avenue is to try to isolate the features that make the above families "reasonable" and others unreasonable, so that only reasonable families can be constructed. That seems somewhat fraught, though.)
As the article notes, these particular infinite families are easy to produce a choice function for: just take the least element of each set. But not all infinite families have enough determined structure for that kind of rule.
If we tried to make it impossible to construct an infinite family of sets, we'd have to disallow relatively reasonable families like the ones I described above. Those are pretty useful, though, so it makes sense to address the problem further downstream.
(I suppose another avenue is to try to isolate the features that make the above families "reasonable" and others unreasonable, so that only reasonable families can be constructed. That seems somewhat fraught, though.)