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Whether one can or can not double the brilliant in real life has nothing to do with the Banach-Tarski paradox. Various mathematical objects are models of various aspects of our universe. $\mathbb R^3$ is typically taken as a model of three-dimensional space. But, it is only a model. Some things that are true in the model are false in space and some that are false in the model are true in space (just like any good model, it is not a precise replica of the thing being modeled, it distorts some things, neglects others, all so that it simplifies the original thing so that it is amenable to be studied by formal means).

The Banach-Tarski paradox is an illustration of (one of) the limitations of $\mathbb R^3$ as a model of the familiar (yet bizarre) ambient space we live in. There are plenty other such incompatibilities. For instance, perfect circles exist in $\mathbb R^3$, you may want to invite your student to construct one in real life. Physics seems to prohibit that just as much it prohibits doubling lumps of gold just by rearranging them piece-wise.

To conclude, models are models. If a model predicts something about whatever it models, then that prediction should be understood within the limitations and distortions made during the modeling process. An incompatibility between the model and reality may entail lots of things. Perhaps a need to abandon the model or a need to adjust it. But sometimes it should just be put away, stored on the shelf of those odd little things that the simplifications assumptions we made bring forth. To your students (or mine, as I often give this example in a first year calculus course), I would say: go ahead and decide how to categorize the Banach-Tarski paradox: is it a serious flaw in the model, a minor disagreement with reality, or an oddity.