Example of how set-theoretic foundations might illuminate applications If possible, please give an example of how set-theoretic mathematical foundations might illuminate an engineer's work.
By engineer, I mean a physical engineer (civil, mining, mechanical, chemical, electrical, etc.) rather than, say, a software engineer.
To clarify: I do not imply that set-theoretic mathematical foundations ought (or ought not) to illuminate an engineer's work, but am merely asking for an example, in case they do happen to illuminate it.
BACKGROUND
The above is my question.  The below merely gives background, if relevant.
I happen to be an electrical engineer with a master's degree in my late 40s.  My professional work is in building construction but my master's study focused on electromagnetics.  Thus, my mathematical applications include vector calculus, special functions, Green's functions, eigenfunctions, contour integrations and so on. I know what an analytic function is in an engineering context, can compute a divergence in parabolic coordinates, and might (with difficulty) exercise differentiable manifolds in the context of surface-borne electric currents; but can barely read logic notation, nor do I have any idea who Galois is or what his theory might be for.
So, recently, to broaden my perspective, I have been reading Richard A. Silverman's 1973 English translation of Georgi E. Shilov's Elementary Real and Complex Analysis. Shilov is an engaging writer and I am enjoying his book, but maybe at my age the brain just grows too inflexible. I believe that there is a point to, say, Cantor's theorem of the uncountable continuum, but I don't seem to be getting the point.
In 1925, Hermann Weyl wrote:

[A set-theoretic approach] contradicts the essence of the continuum, which by its very nature cannot be battered into a set of separated elements.  Not the relationship of an element to a set, but that of a part to a whole should serve as the basis....

Whether Weyl or Cantor is the more right is not for me to say, nor is that the question I am asking; but I can say that Weyl's sentiment makes sense to me. So far, Cantor's sentiments do not make sense to me (unless Cantor's principal sentiment is just that Kant is right and Plato is wrong, in which case I should stop reading Cantorian books, for a merely metaphysical dispute between Cantor and Plato is not a thing I would wish to pursue). Obviously, Cantor's sentiments make sense to a lot of very smart mathematicians, though; so, if there existed an example to bridge Cantor's mental world to the world of an engineer's experience, this would help to motivate my further reading. As matters stand, I am having a hard time understanding how Cantor and friends are talking about anything at all other than abstract games played with arbitrary definitions. So, indeed, an example would help.
Hilbert's grand hotel? Sure, very funny, I suppose: a most ingenious paradox. I should spend the night at Hilbert's grand hotel, sometime. One hears that the hotel's cuisine is transcendental. Meanwhile, I admit that I just don't really get the point. I believe that there is a point; I just haven't gotten it, yet.
To clarify: I am not requesting a general defense of mathematical foundations, but merely a pertinent example, relevant to an engineer, that illuminates the mathematician's interest in mathematical foundations.
 A: The issue here is mathematical precision and reliability.  The example of Green's function that you mentioned is a case in point.  The existence of such a function can be motivated in terms of electric charge equilibrium of a surface, as pointed out by Felix Klein.
Klein notes the distinction between, on the one hand, a physical phenomenon and, on the other, a mathematical theorem attempting to capture the latter. Klein illustrates the distinction by the example that 

in electricity... a conductor under the influence of a charged point is in a state of electrical equilibrium

whose mathematical counterpart is the existence of Green's function.  Klein then concludes:

You see here what is the precise object of these renewed investigations; not any new physical insight, but abstract mathematical argument in itself, on account of the clearness and precision which will thereby be added to our view of the experimental facts. 

(page 245 in Klein, F. "The arithmetizing of mathematics." Bull. Amer. Math. Soc. 2 (1896), no. 8, 241--249)
The issue of precision and reliability became prominent in 19th century mathematics with the emergence of certain paradoxes associated with Fourier series. This led to a realisation that certain issues require more precise justification than what was available up to that point. The combined efforts of Cantor, Dedekind, Weierstrass and others enabled mathematicians to develop more precise justifications, increasing the reliability of their work.
Whether or not an engineer would find all of this relevant depends of course on how intensely he is interested in the reliability of mathematical results.
