[ouid]

"no matter how many you add together", is where this argument breaks down in ZFC. The sphere is indeed the union of all of the singletons consisting of its points, all of which are measure zero. Banach-Tarski is mainly considered "weird" because it describes a partition into so few pieces, and they are rearranged via rigid motions only. It is trivial to come up with bijections between compact finite dimensional manifolds, (https://en.wikipedia.org/wiki/Space-filling_curve). For another example of the axiom of choice wreaking havoc on the notion of measure, see https://en.wikipedia.org/wiki/Vitali_set .

[thaumasiotes]

> "no matter how many you add together", is where this argument breaks down

Interpret it as "adding more points will not necessarily increase the volume, no matter how many points you add". There are plenty of measure-0 sets containing as many points as the continuum does.