Calculus in ordered fields Is there any ordered field smaller that the set of real numbers in which we can do calculus, also with many restrictions ?
If not why ?

I remember reading in Körner's book, "A Companion to Analysis", he discusses problems with calculus over $\mathbb{Q}$, and ordered fields it seems. I think there was a problem with continuity over $\mathbb{Q}$ and of course the fact it's not complete. It might be worth a look if you've access to it. 
Section 1.3
and Chapter 1 anyway it think he deals with ordered fields:
http://books.google.ie/books?id=H3zGTvmtp74C&printsec=frontcover&source=gbs_atb#v=onepage&q&f=false
