You're then discussing math in terms of a set whose properties don't obviously match anything the student has dealt with. It needs to be established, one way or another, that adding infinitesimals etc. is not introducing new behavior.
I don't see why that can't just be stated. Students need somewhere to start from, and there's plenty already that we say "trust us for now, we'll prove it later" - and it's not like the reals actually match anything the student has dealt with, at the corner cases, either...
