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I don't think this answers the question. The fact that there's no bound on $\Vert \hbar D_x u\Vert_{L^2(\mathbb R)}$ in terms of $\Vert xu\Vert_{L^2(\mathbb R)}$ doesn't mean that we can't bound $\Vert \hbar D_x u\Vert_{L^2(\mathbb R)}$ by some other function of $u$.
Do you know anywhere I could read about the fact that the stalk functor reflects finite limits? I've been thinking about it and can't see why it's true (or what's special about the finite case).
I think the construction in the accepted answer there also meets your criteria. The universal construction which maps each group to a cogroup object is a 'relatively natural construction within category theory', and two groups have the same order if and only if their corresponding cogroup objects are isomorphic.
@NoahSchweber I think it's both clunky and nonilluminating. The cardinality proof shows that the entire issue really has nothing to do with approximations, rationals or polynomials. You could extend the definition of 'algebraic' in many ways (for example to the closure of the algebraics under $\exp$ and $\log$) and Cantor's proof would show with no extra effort that there were still reals outside your set, whereas Liouville's proof would fall apart. Cantor's proof uses less to do more, and in a way which is more generalizable.
