On the Splitting Number at Regular Cardinals

TL;DR

This paper investigates the splitting number at regular uncountable cardinals, constructing models where the splitting number equals a given cardinal and relating it to the core model's ordinal under certain assumptions.

Contribution

It constructs a generic extension setting the splitting number at a regular cardinal and establishes a lower bound in the core model assuming the negation of 0^P.

Findings

Constructs models with s(κ) = λ for certain regular cardinals

Shows s(κ) = λ implies o(κ) ≥ λ in the core model under neg 0^P

Relates splitting number to core model ordinal in specific set-theoretic contexts

Abstract

Let $\kappa$,$\lambda$ be regular uncountable cardinals such that $\lambda > \kappa^+$ is not a successor of a singular cardinal of low cofinality. We construct a generic extension with $s(\kappa) = \lambda$ starting from a ground model in which $o(\kappa) = \lambda$ and prove that assuming $\neg 0^{\P}$, $s(\kappa) = \lambda$ implies that $o(\kappa) \geq \lambda$ in the core model.