2026/03/31 by ALBERTO MARCONE, ANDREA VOLPI
paper · doi:10.1017/jsl.2026.10213
crossref issued 2026/05/12 · crossref published 2026/05/12 · crossref published-online 2026/05/12 · crossref created 2026/05/12 · crossref deposited 2026/07/22 · crossref indexed 2026/07/30
Abstract Order dimension theory measures the complexity of partially ordered sets by quantifying how far they are from being linearly ordered. In this article we study classical bounding results for order dimension within the framework of reverse mathematics. We focus on principles asserting that the dimension of a poset can be bounded in terms of the dimension of subposets obtained by removing chains or points, denoted by DBi n \mathsf DBin sans serif upper D upper B i Subscript sans serif n , DBc n \mathsf DBcn sans serif upper D upper B c Subscript sans serif n , and DB p \mathsf DBp sans serif upper D upper B Subscript sans serif p . We prove that, over RCA 0 \mathsf RCA0 sans serif upper R upper C upper A 0 , both DBi n \mathsf DBin sans serif upper D upper B i Subscript sans serif n and DBc n \mathsf DBcn sans serif upper D upper B c Subscript sans serif n are equivalent to WKL 0 \mathsf WKL0 sans serif upper W upper K upper L 0 . To analyze DB p \mathsf DBp sans serif upper D upper B Subscript sans serif p , we introduce a natural strengthening DB p + \mathsf DB+p sans serif upper D upper B Subscript sans serif p Superscript sans serif plus and show that both DB p \mathsf DBp sans serif upper D upper B Subscript sans serif p and DB p + \mathsf DB+p sans serif upper D upper B Subscript sans serif p Superscript sans serif plus are provable from WKL 0 \mathsf WKL0 sans serif upper W upper K upper L 0 and from I Σ 2 0 \mathsf I\boldsymbol Σ 02 sans serif upper I bold upper Sigma 2 Superscript 0 , while B Σ 2 0 \mathsf B \boldsymbol Σ 02 sans serif upper B bold upper Sigma 2 Superscript 0 does not suffice to prove DB p + \mathsf DB+p sans serif upper D upper B Subscript sans serif p Superscript sans serif plus . The latter result is obtained by showing that the statement “ DB p + \mathsf DB+p sans serif upper D upper B Subscript sans serif p Superscript sans serif plus is computably true” is equivalent to I Σ 2 0 \mathsf I\boldsymbol Σ 02 sans serif upper I bold upper Sigma 2 Superscript 0 .