Infinitesimal analysis without the Axiom of Choice

Abstract

It is often claimed that analysis with infinitesimals requires more substantial use of the Axiom of Choice than traditional elementary analysis. The claim is based on the observation that the hyperreals entail the existence of nonprincipal ultrafilters over N, a strong version of the Axiom of Choice, while the real numbers can be constructed in ZF. The axiomatic approach to nonstandard methods refutes this objection. We formulate a theory SPOT in the st-$\in$-language which suffices to carry out infinitesimal arguments, and prove that SPOT is a conservative extension of ZF. Thus the methods of Calculus with infinitesimals are just as effective as those of traditional Calculus. The conclusion extends to large parts of ordinary mathematics and beyond. We also develop a stronger axiomatic system SCOT, conservative over ZF+ADC, which is suitable for handling such features as an infinitesimal approach to the Lebesgue measure. Proofs of the conservativity results combine and extend the methods of forcing developed by Enayat and Spector.