Can Calculus be Formulated with Infinitesimals Without Introducing New Numbers?

[dx]

Is there a formulation of calculus that uses infinitesimals rigorously without introducing an additional number system (non-standard analysis) and without deviating from classical logic?

 

[matt grime]

I would say no, since dx, whatever it is, is not part of any formal system that you will probably accept.

 

[Hurkyl]

Is there a formulation of calculus that uses infinitesimals rigorously without introducing an additional number system

How could this sentence possibly be satisfied? On the one hand, it's asking for new numbers (nonzero infinitessimals), but at the same time you reject introducing new numbers!