I don’t understand what you mean by “the structure behind infinity”
I mean it in contrast to, for example, sqrt(-1). There was clearly a “hole” in polynomial equations: equations that couldn’t be solved. Cardano decided to just define the thing that would fit in that hole and explored the structures that resulted. That turned out fantastically! The structure of complex numbers is incredibly rich and, with 20th century physics, turns out to be arguably more fundamental than the reals.
People tried to repeat that with higher-dimensional numbers. Quaternions have some uses but, as you go up the dimensions, you lose structure, and they become less and less useful.
Infinity strikes me as the same kind of trick as i: there was a hole (“how many natural numbers are there?”) and an object was defined by the shape of that hole. But the results seem to be more like the sedenions (16-dimensional numbers) than complex numbers, and not really worth the bother.
I agree that non-mathematicians can trip over infinities and believe they have found contradictions. This book is very clear that it is defining “paradox” as a surprising result, not a contradiction, and it gives a resolution for each paradox. But having all the results around infinity laid out one after lead me to wonder, what else did infinity give us? Maybe there is something useful that I don’t know about! But I was left with the feeling that such a common concept was actually a dead end.
I think that personal choices about morality are unaffected by the fact that significantly different cultures exist. Perhaps they call for a soupçon more humility, but your moral intuitions remain axiomatic for you.
Rather I think the adjustment needed in some cases is a greater weight on the idea that your moral intuitions are significantly shaped by the culture that you found yourself in, and that the scope of possibilities is wide.
Perhaps this has little practical impact because, though your axioms might be more arbitrary than supposed, you have little choice but to use them. But there will exist people shaped by very different cultures, who formed different rules, and it’s not clear that there’s necessarily any ground for debate; the desire for universal morality might be hopeless.
(Or perhaps communications technology will cause Earth to tend towards a singular culture, giving grounds for a morality universal to all, at least until aliens or disconnected space colonies.)