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?
Answer
To perform nontrivial analysis in a nonstandard extension of $\:\mathbb R\:$ requires much more than simply the existence of infinitesimals. One needs some efficient general way to transfer (first-order) properties from $\:\mathbb R\:$ to the nonstandard extension. This is achieved by a powerful transfer principle in NSA. Lacking such, and other essential properties such as saturation, one faces huge obstacles.
One need only examine earlier approaches to see examples of such problems. For example, see the discussion of the pitfalls of Schmieden and Laugwitz's
calculus of infinitesimals in Dauben's biography of Abe Robinson
(lookup "Laugwitz" in the index) and, for much further detail, see D. Spalt: Curt Schmieden’s Non-standard Analysis, 2001. In short, viewed in terms of
ultrapowers $\mathbb R^\mathbb N,\:$ the S&L approach mods out only by a Frechet filter on $\mathbb N\:$ instead of a free ultrafilter. Thus one loses full transfer of first-order properties, e.g. one obtains only a partially ordered ring, with zero-divisors to boot. Without all of the essential properties of $\mathbb R,\:$ and without a general transfer principle, one obtains a much weaker and much more cumbersome theory as compared to Robinson's NSA.
Another point which deserves emphasis is the role played by logic, in particular the concept of formal languages. One of the major problems with early approaches to infinitesimals is that they lacked rigorous model theoretic techniques. For example, without the notion of a (first-order) formal language
it is impossible to rigorously state what properties
of reals transfer to hyperreals. This logical inadequacy is one of the primary sources of contradictions in the earlier approaches.
Abraham Robinson wrote much on these topics. It is said that he knew more about
Leibniz than anyone. His lofty goal was to vindicate Leibniz'
intuition, and to reverse the historical injustices done to
him by many Whiggish historians. See Abby's collected papers
for much more, and see also Dauben's superb biography of Robinson.
For a brief introduction to the ultraproduct approach to NSA see Wm. Hatcher: Calculus is Algebra, AMM 1982, and Van Osdol: Truth with Respect to an Ultrafilter or How to make Intuition Rigorous. For a much more comprehensive introduction to ultraproducts see Paul Eklof. Ultraproducts for Algebraists, 1977.
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