论文标题
无公理的无限分析
Infinitesimal analysis without the Axiom of Choice
论文作者
论文摘要
人们通常认为，与传统基本分析相比，使用无限量的分析需要更多地使用选择的公理。该主张是基于这样的观察结果：超现实需要在n上存在非首脑超滤波器，这是选择的公理的强大版本，而实数可以在ZF中构造。非标准方法的公理方法驳斥了这一异议。我们在$ - 语言中制定了一个理论地点，足以进行无限的论点，并证明该点是ZF的保守扩展。因此，使用无限量的微积分方法与传统微积分一样有效。结论扩展到了普通数学及其他地区的大部分。我们还开发了更强大的公理系统SCOT，对ZF+ADC保守，它适用于处理Lebesgue度量的无限方法之类的功能。保守性结果的证明结合并扩展了Enayat和Spector开发的强迫方法。
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.
