PEP0238 lament

[Tim Peters]

[TIm]
> I wish the definition of continuity were easier to teach too, but that
> doesn't mean I'll settle for an easier definition that doesn't actually
> work <wink>.

[Moshe Zadka]
> Here is one that is easier to teach, if you just reverse the order
> of teaching calculus and first-order logic and basic set theory.
>
> A function is continuous at c if, when embedding the world of Reals
> into a saturated model (or, say, 2^{Beth_omega} saturated, if you
> don't want to get into subtle assumptions), for every infinitesimal x,
> f(c+x)-f(c) is an infintesimal.
>
> Of course, you might think that teaching about saturation is hard --
> well, it might be ;-)

Na, they always leave *some* of the foundations in intro calculus courses
fuzzy.  H. Jerome Keisler wrote a very good intro calculus text based on
non-standard analysis, unfortunately now out of print.  See

    http://www.math.wisc.edu/~keisler/books.html

for publication details.  He didn't try to explain the intricacies of model
theory, he just spelled out "the rules" for working with infinitesimals and
got on with it.  Huge steaming masses of epsilon-delta proofs were
conspicuous by absence, so he was able to get to "the interesting stuff" a
lot quicker that way.  A good feel for the flavor of the approach can be
gotten by skipping to the end ("Part IV: Turning Calculus into Algebra") of

    http://online.sfsu.edu/~brian271/nsa.pdf

and reading "~=" as "infinitely close to".

> i-always-blamed-cauchy-and-weirstrass-for-tormenting-the-world-
> with-epsilons-and-deltas-ly y'rs, Z.

i-personally-blame-guido-for-letting-it-stand<wink>-ly y'rs  - tim