Linear extensions of partial orders and reverse mathematics
Emanuele Frittaion, Alberto Marcone · Mathematical logic quarterly · 2012
Abstract We introduce the notion of τ‐like partial order, where τ is one of the linear order types ω, ω*, ω + ω*, and ζ. For example, being ω‐like means that every element has finitely many predecessors, while being ζ‐like means that every interval is finite. We consider statements of the form “any τ‐like partial order has a τ‐like linear extension” and “any τ‐like partial order is embeddable into τ” (when τ is ζ this result appears to be new). Working in the framework of reverse mathematics, we show that these statements are equivalent either to \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {B}{\Sigma }^{0}_{2}$\end{document} or to \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {ACA}_0$\end{document} over the usual base system \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {RCA}_0$\end{document} .