论文标题
表弟的引理的相反数学
The reverse mathematics of Cousin's lemma
论文作者
论文摘要
表弟的引理是一种紧凑的原理，在研究仪表积分（Lebesgue积分的概括）时自然会产生。我们使用弗里德曼（Friedman）和辛普森（Simpson）在二阶算术中使用弗里德曼（Friedman）和辛普森（Simpson）的反向数学研究表弟的引理的公理强度。我们证明，超过$ \ mathsf {rca} _0 $： （i）堂兄的连续功能的引理相当于系统$ \ mathsf {wkl} _0 $; （ii）Cousin的Baire 1功能的引理至少与$ \ Mathsf {aca} _0 $一样强； （iii）Cousin的Baire 2函数的引理至少与$ \ MATHSF {ATR} _0 $一样强。
Cousin's lemma is a compactness principle that naturally arises when studying the gauge integral, a generalisation of the Lebesgue integral. We study the axiomatic strength of Cousin's lemma for various classes of functions, using Friedman and Simpson's reverse mathematics in second-order arithmetic. We prove that, over $\mathsf{RCA}_0$: (i) Cousin's lemma for continuous functions is equivalent to the system $\mathsf{WKL}_0$; (ii) Cousin's lemma for Baire 1 functions is at least as strong as $\mathsf{ACA}_0$; (iii) Cousin's lemma for Baire 2 functions is at least as strong as $\mathsf{ATR}_0$.
