analysis over non-Archimedean ordered fields Can anyone suggest any good references for (or any experts on) analysis over non-Archimedean ordered fields, such as the field of rational functions in one variable (ordered at 0, or if you prefer at infinity)?
It may not be entirely clear what I mean by the foregoing, so here's the kind of thing I mean: Take every definition in real analysis that refers to the field of reals and replace it by a relativized version in which quantification over the reals is replaced by quantification over $K$, where $K$ can be any ordered field you like.  Now take every theorem in real analysis and ask whether it's true relative to $K$.  (E.g., the Intermediate Value Theorem is false for "calculus over the rationals".)   
It's fairly easy to think about what's true over $K$ when $K$ is Archimedean, since every Archimedean ordered field is a subfield of the reals, but it's harder for me to think about the non-Archimedean case (even simple cases like the field of rational functions in one variable or the field of formal Laurent series).  I suspect someone has already worked all this out, but I'm not sure where to look.
 A: Well, there seems to be a lot of literature. I have encountered similar questions once when discussing problems in deformation quantization. here the ordered field is simply the field of formal Laurent series $\mathbb{R}((\hbar))$ with the ordering that $\hbar$ is positive (this fixed it uniquely). The idea was to transfer as much of stuff from $C^*$-algebra theory to the formal star products which are algebras of the above non-Archimedian field. Ultimately, we wanted to develop a spectral theory for these algebras, which utterly failed :) Nevertheless, you might be interested in examples of such fields and the "completed Newton Puiseux field" might be a funny one. First you take the algebraic or real closure of the formal Laurent series yielding the Newton-Puiseux series. This is not yet complete (in the sense of convergent Cauchy sequences) so you still have to complete it. The result is again field (either real, i.e. ordered) or algebraically closed, the smallest one containing the formal Laurent series.
Let me probably clarify what I mean by completion here: since you have an ordered field, the naive $\epsilon$-definition of a Cauchy sequence makes sense with the main point that the $\epsilon$ is now a positive element of you field and not just a positive real number. So this defines a topology on your ordered field, but also a uniform structure. So one can speak about Cauchy sequences and completeness etc. It is then a standard argument to show that the completion (in the sense of uniform structures, i.e. taking Cauchy sequences module zero sequences) gives again an ordered field and the original one is included via constant sequences as usual. The slightly less trivial point is that if you start with an ordered field which is real closed then the completion is again real closed. This is needed to see that the completed Newton-Puiseux series are indeed both: real (or algebraically, in the case you started with $\mathbb{C}$ instead of $\mathbb{R}$) closed and (Cauchy) complete.
A first reading may be
Narici, N., Beckenstein, E., Bachman, G.: Functional Analysis and Valuation Theory. New York: Marcel Dekker, 1971.
However, the main point is that analysis does not work too well. The reasons is that whenever you need suprema, you're on your own. And this happens quite often in analysis :) The other problem arising is that $1/n$ is no longer a zero sequence, a fact which is also used very often in analysis: you named already the intermediate value theorem...
So my conclusion is that notions of positivity work well (needed a lot in formal DQ and representation theory of $^*$-algebras) but notions of calculus do not work well.
A: There is a well-developed analysis over the Levi-Civita ordered field consisting of functions from $\Bbb Q$ to $\Bbb R$ with left-finite supports. The main authors are M. Berz and K. Shamseddine; see, for example, http://www.uwec.edu/surepam/media/RS-Overview.pdf 
I would like to mention also a general monograph on ordered algebraic structures containing rich material of more algebraic nature, the book by S. Priess-Crampe, Angeordnete Strukturen: Gruppen, K\"orper, projektive Ebenen, Springer, 1983.
These activities should not be confused with a more well-known non-Archimedean analysis over local fields and their extensions (treated in the book by Narici et al, and many others). These fields have no order relations agreed with their natural structures, though some positivity and monotonicity notions can be useful also in this framework (see "Ultrametric calculus" by W.H. Schikhof). 
A: I don't understand Stefan's suggestion that you can get a field by taking (presumably) the Dedekind-Macneille completion of the ordered field of Newton-Puiseux series or any non-archimedean field. A non-archimedean field cannot be order-theoretically complete, e.g., because the set of integers is bounded above but can have no least upper bound (if $x > \mathbb{Z}$, then so is $x - 1$).
However, many notions of analysis do make sense over a real closed field. Such fields can be non-archimedean, the above mentioned field of Newton-Puiseux series being a classic example. This is part of the subject matter of real algebraic geometry (see the book of that name by Bochnak, Coste and  Roy), or more generally of tame topology (see the book Tame Topology and O-minimal Structures by  van den Dries). There are also lots of papers and survey articles on these topics at the Real Algebraic and Analytic Geometry preprint server: http://www.maths.manchester.ac.uk/raag/.
A: I am not a big expert in this, but it sounds as if you might check out books on non-standard analysis. For you come quite close to a setting when you work in a "fragment" of non-standard analysis framework, as being done in some parts of real algebraic (and analytic) geometry, when the one works over a real closed field obtained (say) by adjoining a positive infinitesimal to a real closed field, i.e. turing it into a field of Puiseux series.
A: Foundations of Analysis over Surreal Number Fields by Norman Alling may be along the lines of what you're looking for. For example, it is shown that an Implicit Function Theorem holds over the Surreals.
In Philip Ehrlich's paper "The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small” (2012) it is shown that all real-closed ordered fields underlying the hyperreal number systems (i.e. the nonstandard models of analysis) are isomorphic to initial subfields of the Surreal numbers, thus all initial subfields of the surreal numbers (and the Surreals themselves) admit relational extensions to a model of nonstandard analysis in which the transfer principle holds.  For more, see Philip's excellent answer here.
