Extraterrestrials and Calculus

One of the odder parts of the background for David Brin's Uplift series is that the Galactic civilization does not use calculus. They regard it as as something from a primitive people. According to the Niss machine in Infinity's Shore:

The result? A mix of the brilliant and the inane. Abstract differential analysis and cabalistic numerology. Algebra, astrology, and geometrical topology. Much of this amalgam was based on patently absurd concepts, such as continuity, or aptly named irrational numbering, or the astonishing notion that there are layered infinities of the divisibly small.

On the other hand, there are mathematical results about plain ordinary finite numbers that cannot be proved using finite number theory. Presumably, the Galactics would not know those theorems.

On the gripping hand, Goodstein's Theorem and similar results can be proved using second-order number theory. Second-order number theory also turns out to be as strong as the complex number system when exponentiation and complex conjugation are included. So if you object to Cantorian set theory but accept complex numbers, you can still prove Goodstein's Theorem, etc.

Devising such a proof will be left as an exercise for the reader.