Year: 2024
Type: article
Abstract: In a recent paper, Cummings, Eisworth and Moore gave a novel construction of minimal non- $$\sigma $$ σ -scattered linear orders of arbitrarily large successor size. It remained open whether it is possible to construct these orders at other cardinals. Here, it is proved that in Gödel’s constructible universe, these orders exist at any regular uncountable cardinal $$\kappa $$ κ that is not weakly compact. In fact, for any cardinal $$\kappa $$ κ as above we obtain $$2^\kappa $$ 2 κ many such orders which are pairwise non-embeddable. At the level of $$\aleph _1$$ ℵ 1 , their work answered an old question of Baumgartner by constructing from $$\diamondsuit $$ ♢ a minimal Aronszajn line that is not Souslin. Our uniform construction is based on the Brodsky–Rinot proxy principle which at the level of $$\aleph _1$$ ℵ 1 is strictly weaker than $$\diamondsuit $$ ♢ .
Source: European Journal of Mathematics
Author: Roy Shalev
Institution: Bar-Ilan University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
Award: G3416096842
OpenAlex ID: W4404920675