TITLE: Why adjoining non-Archimedean element doesn't work as calculus foundation?
QUESTION [10 upvotes]: Consider the smallest ordered field that contains R and does not satisfy the Archimedean property. I assume this is a much simpler construction than ultrafilters and other big caliber artillery used in non-standard analysis. Why does this approach fail?

REPLY [2 votes]: If the assumption of the question is that using a simple-minded non-Archimedean field cannot be used to get interesting mathematical results, that assumption can be challenged.  For example, Dehn used a simple-minded non-Archimedean field to resolve one of Hilbert's famous problems; see here.