Dominating numbers at singular cardinals

2025/08/16 by Yusuke Hayashi, Hayashi, Yusuke
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis

paper · pdf · doi:10.48550/arxiv.2508.12018

openalex publication_date 2025/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the generalized dominating number \mathfrakd_μ at a singular cardinal μ of cofinality κ. We show two lower bounds: in ZFC, cf([μ]^κ,⊆) ≤ \mathfrakd_μ, and under mild cardinal-arithmetic assumptions, 2^<μ ≤ \mathfrakd_μ. We also clarify when \mathfrakd_μ can differ from 2^μ: assuming GCH and κ= cf(μ) > ω, a finite-support iteration of Cohen forcing of length μ⁺⁺ yields \mathfrakd_μ < 2^μ. On the other hand, for κ= cf(μ) = ω, natural μ-cc posets force \mathfrakd_μ = 2^μ.

Citations

• Cichon's Diagram for uncountable cardinals

Related

• A Lower Bound for Generalized Dominating Numbers
• On CON(\mathfrak d_λ> cov_λ(meagre))
• On CON(Dominating_lambda > cov_λ(meagre))
• Blowing up the power of a singular cardinal of uncountable cofinality with collapses
• Higher Dimensional Cardinal Characteristics for Sets of Functions II