I tried teaching elementary calculus a couple of times using Keisler's book
as a basis. I never tried to bring up model theory in these courses, but
as you say, asked the students to accept some things without proof,
sometimes tacitly (no one brought up certain existence questions, so I
didn't either). One big difficulty I found was that students who were
using calculus in science or engineering courses were not able to relate a
calculus based on nonstandard analysis to the calculus in their other
courses. You might think that the way some physicists like to talk about
infinitesimals as infinitely small quantities, or the like, they would be
happy to have mathematicians give some sort of ontological status to
infinitely small quantities, but most of them seem not to have ever wanted
there to be positive numbers which are smaller than every positive number
of the sort they use in measurements. In any case, as you suggest, a lot
of calculus teachers resisted nonstandard calculus, usually on the grounds
that nonstandard calculus was more complicated than it had to be. I came
to sympathize with such people after some years of contemplating
nonstandard analysis. I can sum up my final view by saying that I came to
believe that a complete linearly ordered (therefore Archimedean) field,
augmented with some limit theory in the spirit of Cauchy and Weierstrass
was simpler (that word "simple" again!) than using a non-Archimedean field,
and also easier to relate to applications, especially in connection with
the sort of approximations one works with when using computers.
>Anyway, to teach about infinitesimals does not really
>require model theory. If you don't insist on carrying
>out Leibniz's full program, and you just want to give
>some examples of infinitesimals and illustrate their
>properties, any non-Archimedean ordered field will do.
>One such field, quite easy to describe, is the
>field of all rational functions (with real coefficients,
>with one variable, with the convention that two
>rational functions are considered to be the same
>if they differ at only finitely many points of R).
>For the ordering, say that p is greater than q if
>there exists a real number t such that, whenever
>s > t, then p(s) > q(s). You can state this fact
>without proof, if you're teaching to undergraduates:
>It can be shown that if p and q are any rational
>functions, then either they are the same (in the
>sense above) or one is greater than the other
>(in the sense above). Hence the rational functions
>form a chain-ordered field.
>
Yes, but I didn't find this level of abstraction workable in elementary
calculus courses of the sort I taught in a number of different
universities. I used to use examples of this sort for various purposes in
real variable courses.
>In any ordered field, an infinitesimal is a number
>that lies between 1/n and -1/n for all positive
>integers n. An ordered field is Archimedean if
>and only if 0 is the only infinitesimal. In
>the field of rational functions, if x is the variable,
>then 1/x is an infinitesimal.
>
>This gives us an easy example of several other
>equivalent definitions of Archimedean, as well as
>an interesting example of a field that is not
>Dedekind-complete.
>
>When I taught about fields this semester, I gave
>only a few examples. I included an example of
>a finite field (5 elements) because I felt that
>that emphasizes that the abstract axioms may be
>satisfied by something that doesn't even remotely
>resemble familiar things.
>
>Well, you did ask. :)
>
>
>
>************************************************************
>Eric Schechter *** http://www.math.vanderbilt.edu/~schectex/
>************************************************************
>
Perhaps your history of mathematics course is for upper undergraduates who
have a fairly good grounding in algebra and analysis? Also, do you do
anything with infinitely large numbers, as reciprocals of infinitesals?
Gordon Fisher gfisher@shentel.net