I prefer to use a proper subset of Robinson’s hyperreals (his R*) only requires the cofinite (Fréchet) filter on N. Call my subset R†, the set of ratios of real polynomials (the familiar “rational functions”) of the index variable, where two pairs of polynomials have the same ratio (belong to the same equivalence class) if the set of values of the index for which the ratios are not equal is finite.

This is the classic example of a non-Archimedean ordered set, and, if statements about polynomial ratios are considered true only the same conditions for equality hold, the transfer principle holds for all first-order statements. The proof requires only the properties of filters in general, and not the property of an ultrafilter that EVERY set of natural numbers or its complement has to be a member of the ultrafilter.

I am writing an appendix for Thompson’s _Calculus_made_Easy_ that uses my R† to justify the infinitesimals.

The great triumph of standard (real number) math is Weierstrass epsilon-delta definition of a limit, which tells one when one has found a limit (using only the real numbers), but leaves no clue as to how to find it.

In effect, nonstandard methods allow one to calculate beyond the infinite decimal precision of standard reals, and round off directly to the nearest standard real, the limit sought, Robinson’s “standard part”, without all that tedious mucking about in shrinking deltas and epsilons.

Don’t you mean “tangent vectors to N” instead of R?

The tangent space at decomposes into the sum of the tangent space to M at x, and the tangent space to the cotangent space at x.

Don’t we need a connection for this?