Anyone know much about this subject? I'm satisfied with my understanding of the basics of the transfer principle. My algebra text has a chapter on real closed fields, including a section on Tarski's theorem, and I see how it relates to the transfer principle. A paper I read recently, plus a revelation I had, cleared up what was the thing bugging me the most about the decidability of real arithmetic. However, I am now more troubled by other things. One of the key things about why real arithmetic is complete is because real arithmetic doesn't know what "integer" means. Yet, the hyperintegers are used in NSA! Also, the completeness of real arithmetic works against analysis; the algebraic numbers are also a model of the theory of real arithmetic, and transcendental functions cannot exist among the algebraic numbers... thus they also must not be a part of real arithmetic. So, clearly, this simle theory is not sufficient to do nonstandard analysis. I have found a little bit of information on superstructures... not nearly enough for me to feel I really understand it. Does anyone know of any good online resources for NSA? If not, maybe a good book or two?