Does the mathematics of physics require impure set theory? Suppose for the sake of this question that all mathematics is ultimately reducible to set theory in such a way that the only mathematical objects there really are, are sets.
Now, there is a common distinction between pure and impure set theory. Pure set theory is built up from the empty set and only involves "pure" sets--- sets that only contain other sets as members. Impure set theory, by contrast, involves urelements--- non-set individuals which can be members of sets.
My question is whether the mathematics needed for physics (assuming it all gets cashed out in terms of set theory) is of the pure or impure variety? Are there primitive relations of physics (where a primitive relation of physics is something unique to physics--- not something like equality/identity) that hold between mathematical entities (like numbers) and physical entities? If so, could you provide examples?
 A: Impure set theory can be interpreted in terms of pure set theory, so the question is moot.
You may ask which interpretation is "more natural". But embedding arithmetic, analysis, and calculus in terms of set theory is already a fairly convoluted and unnatural thing.
A: I'm not entirely sure this question is answerable in terms of physics.
The reason is that all physics theories are reducible to combinations of data structures (e.g. scalars, vectors, matrices, tensors) and associated behaviors. Data structures and associated behaviors are in turn (and pretty much by definition) entities that can be represented within any formal system that has enough richness and complexity to build a Turing machine within it. Even odd physics issues such as quantum entanglement can be modeled in this fashion, just very inefficiently.
So, the problem is that pure and impure set theories are both rich enough (easily!) to build Turing machines. Either one could be used represent any formal physics theory. Since pure sets are a bit simpler, that would (I guess) be the winner between the two options.
However, your real question may be more along the lines of asking if there are any unique, atomic, indivisible things or properties in physics that match better with urelements than with pure sets. Asked that way, I'd say there is a pretty good chance that there are. Physics hasn't quite found them yet, but structure does get simpler as you get smaller. For example, nothing in physics seems to get below $1/3$ of an electron charge or less than $1/2$ of a unit of spin, so something firmly "atomic" or "indivisible" may be going on with those units.
However, I would also point out that such smallest-units in physics tend to come in mutually annihilating pairs, not as simple additions. The idea of mutually annihilating pairs is not typically seen in the all-positive-additions way in which sets are typically constructed, or at least not something I've seen in set theory.
So, if you want to broaden the question a bit and say "what kinds of mathematical systems best seem to capture the natural structure of physics as we observe it?", I would say "something that is based on the creation and annihilation, at many different levels, of mutually annihilating pairs of properties or entities." It would be akin perhaps to a more discrete version of the framework found in quantum field theory.
I am not specifically aware of any specific formal system of that type in mathematics, at least not as part of the search for more fundamental mathematical building blocks like sets.
