No axiom of choice, but most working mathematicians reject constructivism.

Plus there are some pretty big differences. Do you really want to explain to students why you don't consider f(x) = 1 if x >= 1 and 0 otherwise to be a valid definition of a function? And (in some ways more importantly) why most of the rest of the world does? Furthermore the intermediate value theorem is pretty useful and I wouldn't want to lose it.