Someone with my mathematicians' dna finds it incomprehensible that anyone could be happy studying a subject in which one did not know what all the words mean. I myself was never content until after sitting through, as a junior or sophomore, a course in which George Mackey laid out the foundations of analysis (for undergraduates), beginning with only the axiom that there exists at least one infinite set, proceeding to the Peano postulates, and definition of the natural numbers as I outlined above, and then the rationals and reals, fundamentals of metric spaces, and the proofs of the basic theorems of calculus. At the end I felt and thought: ahhh...., at last I can be at peace with these previously fuzzy statements! I still wonder how anyone can think they know anything about calculus who cannot define, or at least characterize by axioms, the real numbers. If you don't know what a real number is, why bother to state theorems about them, e.g. what do you mean when you invoke the intermediate value theorem? Of course the student is not at fault, since they have been taught by a teacher who did not bother to explain what was going on. The rubber hits the road with a student who thinks for instance that all real numbers are integers, or at best that they are all finite decimals that can be displayed on their calculator. With such an understanding, of course all the theorems of calculus are actually false!