There are models of differential geometry in which the intermediate value theorem is not true but every function is smooth. In fact I have a book sitting on my desk called "Models for Smooth Infinitesimal Analysis" by Ieke Moerdijk and Gonzalo E. Reyes in which the actual construction of such models is carried out. I'm quite new to this entire subject and I only stumbled upon it because I was trying to find something like non-standard analysis for differential geometry.

Already I'm liking the more natural formulations for differentials and tangent vectors in the new setting although I can see that true mastery of all the intricacies will require more background in category theory like Grothendieck topologies. So my questions are a bit philosophical. Suppose some big conjecture is refuted in one of these models but proven to be true in the classical setting then what exactly would that mean for classical differential geometry? Is such a state of affairs possible or am I missing something that rules out such a possibility like a metatheorem that says anything that can be proven in the new models can be proven in the usual classical model? More specifically what is the exact relationship between the new models and the classical one? Could one even make any non-trivial comparisons? References for such discussions are welcome. I'm asking the question here because I suspect there might be some experts familiar with synthetic differential geometry that will be able to illuminate the connection to the classical theory.

**Edit**: Found a very lively and interesting discussion by John Baez, Andrew Stacy, Urs Schreiber, Tom Leinster and many others on n-category cafe called Comparative Smootheology although I couldn't make out the exact relation to SDG.

a = eb for all e, then a = b. Classically, you can show the only infinitesmal is 0, which makes the cancellation axiom inconsistent. (As it happens, this is essentially Berkeley's critique of calculus, which is what led to the development of the limit concept!) $\endgroup$