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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.

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What I'd like to know, if anyone can answer it, is why couldn't they just do synthetic differential topology in a classical-style topos? –  Harry Gindi Dec 11 '09 at 10:50
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Nilpotent infinitesmals are not compatible with the excluded middle. For example, in smooth infinitesmal analysis you posit the existence of a family of infinitesmals e such that e^2 = 0, and the cancellation axiom, which says that if ea = 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!) –  Neel Krishnaswami Dec 11 '09 at 17:12
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The "comparative smootheology" discussion is about less-dramatic generalizations of manifolds than that considered in SDG. For example, you don't have nilpotent infinitesimals. –  arsmath Dec 27 '10 at 22:27

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Perhaps I can make the implications of what Harry said a bit more explicit. A well-adapted model of SDG embeds smooths manifolds fully and faithfully. This in particualar means that the SDG model and the smooth manifolds "believe" in the same smooths maps between smooth manifolds (but SDG model contains generalized spaces which do no correspond to any manifold), and moreover precisely the same equations hold in the SDG model and in smooth manifolds. In this sense SDG is conservative: the model will never prove an invalid equation involving smooth maps between smooth manifolds.

The situation is really quite similar to other situations where we have to distinguish between truth and meaning inside a model and truth and meaning outside the model. For example, there are models of set theory which violate the axiom of choice, but these models are built in a setting where the axiom of choice holds. This is no mystery or magic, as long as we remember that a statement can have a different meaning inside the model than outside. The same applies to SDG: when the internal meaning of statements in the model is appropriately interpreted on the outside, nothing can go wrong (that's what a model is, after all).

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From n-lab:

"A topos T modelling the axioms of synthetic differential geometry is called (well) adapted if the ordinary differential geometry of manifolds embeds into it, in particular if there is a full and faithful functor Diff →T from the category of ordinary smooth manifolds into T."

The main point here is that we can develop the theory abstractly, then apply the theory to the category of differentiable manifolds, which is equivalent via this functor to a subcategory of this nice topos where we're working. In particular, all of the abstract results restrict to the classical results.

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