You might be interested in the book A Primer of Infinitesimal Analysis by John L. Bell, which is another approach to infinitesimals that uses intuitionistic logic.

https://www.amazon.com/Primer-Infinitesimal-Analysis-John-Be...

Any interesting property of this logic and model is that all functions are infinitely differentiable.

Other introductions are An Invitation to Smooth Infinitesimal Analysis by John L. Bell and Synthetic Differential Geometry by Michael Shulman.

http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf

http://home.sandiego.edu/~shulman/papers/sdg-pizza-seminar.p...

This kind of approach simply doesn't make sense

It was poorly explained there, but essentially that style of reasoning does work: http://www.amazon.com/Primer-Infinitesimal-Analysis-John-Bel...

And I at least find that approach easier and more useful. (I learned it from the Feynman lectures on physics, where he didn't axiomatize it; the above link does.)