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Is the theory of partial differential fields a first-order or higher-order logic theory? Is the theory of differential fields already equipped with partial derivatives? I was told that the General Theory of Relativity is at least a second-order logic theory, because it involves solutions of partial derivatives -- is this true? If so, does that mean Maxwell's Equations are also second-order at least as it requires calculating partial derivatives as well?

Thanks!

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    Well, the theory of topological spaces is a second-order theory, so anything to do with continuity already must be second-order. The theory of manifolds is higher-order still with its atlases and charts. But what does it matter? We can just embed it all inside set theory, which is first-order.2011-09-08
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    Right. So if the theory of topological spaces is second-order but can be embedded in ZFC, then isn't it first-order then?2011-09-08
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    First-order axiomatizations of the notion of differentiaal field can and do yield useful information. It does not matter that the objects the theory deals with are "really" functions.2011-09-08
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    On the topic of axiomatising special and general relativity, [this paper](http://dx.doi.org/10.1007/978-1-4020-5587-4_11) gives first-order axiomatisations for both and detailed studies of their models, including a completeness theorem.2011-09-08
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    @ZhenLin Please consider converting your comments into an answer, so that this question gets removed from the [unanswered tab](http://meta.math.stackexchange.com/q/3138). If you do so, it is helpful to post it to [this chat room](http://chat.stackexchange.com/rooms/9141) to make people aware of it (and attract some upvotes).2013-10-10

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