论文标题

最大的五个反向数学

The Biggest Five of Reverse Mathematics

论文作者

Normann, Dag, Sanders, Sam

论文摘要

反向数学的目的(简称RM)是找到证明给定数学定理所需的最小公理。这些最小公理几乎总是等同于定理,它在RM的基础理论上起作用,RM的基础理论是一种弱的可计算数学系统。 RM的五大现象是观察到,普通数学的大量定理可以在基本理论中证明,或者等同于仅四个系统之一。这五个系统一起称为“五巨头”。本文的目的是极大地扩展了五大现象,如下所示:据说有两种根本不同的RM方法,其中主要区别是该语言是否仅限于二阶对象,还是一个允许三阶对象。在本文中,我们通过一方面建立涉及二阶Big五个系统的众多等效性来团结RM的这两条链,另一方面是(可能)不连续的函数的分析,涉及二阶Big五个系统以及众所周知的三阶定理。我们都研究了相对驯服的概念,例如Cadlag或Baire 1,以及潜在的野性概念,例如准官方。我们还表明,上述三阶定理的轻微概括和变化远远超出了五巨头。

The aim of Reverse Mathematics(RM for short)is to find the minimal axioms needed to prove a given theorem of ordinary mathematics. These minimal axioms are almost always equivalent to the theorem, working over the base theory of RM, a weak system of computable mathematics. The Big Five phenomenon of RM is the observation that a large number of theorems from ordinary mathematics are either provable in the base theory or equivalent to one of only four systems; these five systems together are called the 'Big Five'. The aim of this paper is to greatly extend the Big Five phenomenon as follows: there are two supposedly fundamentally different approaches to RM where the main difference is whether the language is restricted to second-order objects or if one allows third-order objects. In this paper, we unite these two strands of RM by establishing numerous equivalences involving the second-order Big Five systems on one hand, and well-known third-order theorems from analysis about (possibly) discontinuous functions on the other hand. We both study relatively tame notions, like cadlag or Baire 1, and potentially wild ones, like quasi-continuity. We also show that slight generalisations and variations of the aforementioned third-order theorems fall far outside of the Big Five.

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