I like to think of this as a game, with one player choosing the axioms and the other choosing a model. If the first player picks a (countable) set of axioms, th

I like to think of this as a game, with one player choosing the axioms and the other choosing a model. If the first player picks a (countable) set of axioms, the second player can always respond with a countable model. Likewise, if the second player picks a countable model, the first player can always extend the axioms in a consistent way, to rule out that model. This can alternate back-and-forth forever. Uncountability is hence a 'leaky abstraction': something we want to investigate and study in general terms, even though particular occurances might have some loophole/edge-case. I think about infinity and infinitesimals in a similar way, like iterative processes (e.g. the natural numbers arise from a process that increments; calculus arises from iteratively shrinking 'dx', e.g. by halving; etc.). Combining/interleaving such processes is tricky, so it's often more convenient to take their limits individually and manipulate those as objects; that's justified if those manipulations could potentially be implemented by some interleaving, but can otherwise result in paradoxes (e.g. Thomson's lamp)