2023/09/07 by Assaf Rinot, Rinot, Assaf, Shira Yadai +3 · 1 citation
Mathematics · #03E55 #Advanced Topology and Set Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #Primary 03E05. Secondary 03E35
paper · pdf · doi:10.48550/arxiv.2309.03821
openalex publication_date 2023/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We initiate the study of the spectrum Vspec(κ) of sets that can be realized as the vanishing levels V(T) of a normal κ-tree T. The latter is an invariant in the sense that if T and T' are club-isomorphic, then the symmetric difference of V(T) and V(T') is nonstationary. Additional features of this invariant imply that Vspec(κ) is closed under finite unions and intersections. The set V(T) must be stationary for an homogeneous normal κ-Aronszajn tree T, and if there exists a special κ-Aronszajn tree, then there exists one T that is homogeneous and satisfies V(T)=κ (modulo clubs). It is consistent (from large cardinals) that there is an ℵ2-Souslin tree, and yet V(T) is co-stationary for every ℵ2-tree \mathbf T. Both V(T)=∅ and V(T)=κ (modulo clubs) are shown to be feasible using κ-Souslin trees even at some large cardinal close to a weakly compact. It is also possible to have a family of 2κ many κ-Souslin trees for which the corresponding family of vanishing levels forms an antichain modulo clubs.