Foundations of Mathematics

I watched this interesting 2017 PBS video "Crisis in the Foundation of Mathematics" today. I was struck by how in the Zermelo-Fraenkel set theory of the foundations of mathematics, the existence of infinite series has to be defined as an axiom: it's not automatic that they exist.

Another thing that came to mind is the weird way that mathematicians try to formulate a foundation for their discipline, for example, in order to broaden the natural (counting) numbers into the integers (that is, to include the negative numbers), they define an integer as the difference between two natural numbers. What I don't understand is why they don't just define negative numbers as an abstraction that incorporates the "take away" operation (subtraction) into the number. Imaginary numbers would then be a kind of "half-way" take-away, what amounts to a ninety-degree rotation (half of 180°).

I say "I don't understand," but seriously: why should anyone expect me to understand? My training is in physics, not mathematics, after all.