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 $\mathbb{N}$, 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-∈-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 $\mathbf{ZF}+\mathbf{ADC}$, 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.

通常认为，与传统的基本分析相比，使用无穷小进行分析需要更多地使用“选择公理”。该主张基于以下观察：超真实感导致存在非主超滤器。$\mathbb{ñ}$，是“选择公理”的强大版本，而实数可以在ZF中构建。非标准方法的公理化方法反驳了这一反对意见。我们用st -∈语言建立了一个理论SPOT，足以进行无穷小论证，并证明SPOT是ZF的保守扩展。因此，具有无穷小数的微积分方法与传统微积分方法一样有效。该结论扩展到普通数学的大部分及以后。我们还开发了一个更强的公理化体系SCOT，保守了$\mathbf{采埃孚}+\mathbf{ADC}$，适用于处理诸如Lebesgue度量的无穷小方法之类的功能。保守性结果的证明结合并扩展了Enayat和Spector开发的强制方法。