2018/08/27 by Sam Sanders, Sanders, Sam · 1 citation
Computer Science · Mathematics · #03B30 #03D65 #03F35 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis #math.LO #msc:03B30 #msc:03D65 #msc:03F35
paper · pdf · doi:10.48550/arxiv.1808.08785
19 pages, one figure. To appear in Notre Dame Journal for Formal Logic
openalex publication_date 2018/08/27 · openalex created_date 2018/08/31 · arxiv created 2020/05/28 · arxiv updated 2020/05/29 · openalex updated_date 2026/07/28
Reverse Mathematics (RM hereafter) is a program in the foundations of mathematics founded by Friedman and developed extensively by Simpson and others. The aim of RM is to find the minimal axioms needed to prove a theorem of ordinary, i.e. non-set-theoretic, mathematics. As suggested by the title, this paper deals with the study of the topological notions of dimension and paracompactness, inside Kohlenbach's higher-order RM. As to splittings, there are some examples in RM of theorems A, B, C such that A↔(B\wedge C), i.e. A can be split into two independent (fairly natural) parts B and C, and the aforementioned topological notions give rise to a number of splittings involving highly natural A, B, C. Nonetheless, the higher-order picture is markedly different from the second-one: in terms of comprehension axioms, the proof in higher-order RM of e.g. the paracompactness of the unit interval requires full second-order arithmetic, while the second-order/countable version of paracompactness of the unit interval is provable in the base theory of second-order RM. We obtain similarly 'exceptional' results for the Urysohn identity, the Lindelöf lemma, and partitions of unity. We show that our results exhibit a certain robustness, in that they do not depend on the exact definition of cover, even in the absence of the axiom of choice.