Non-linear iterations and higher splitting

Abstract

We show that generalized eventually narrow sequences on a strongly inaccessible cardinal $\kappa$ are preserved under the Cummings-Shaleh non-linear iterations of the higher Hechler forcing on $\kappa$. Moreover assuming GCH, ${\kappa }^{<\kappa }=\kappa$, we show that: (1) if $\kappa$ is strongly unfoldable, ${\kappa }^{+}\le \beta =cf(\beta )\le cf(\delta )\le \delta \le \mu$ and $cf(\mu )>\kappa$,then there is a cardinal preserving generic extension in which $$\mathfrak{s}(\kappa )={\kappa }^{+}\le \mathfrak{b}(\kappa )=\beta \le \mathfrak{d}(\kappa )=\delta \le 2^{\kappa }=\mu .$$ (2) if $\kappa$ is strongly inaccessible, $\lambda >{\kappa }^{+}$, then in the generic extension obtained as the $<\kappa$-support iteration of $\kappa$-Hechler forcing of length $\lambda$ there are no $\kappa$-towers of length $\lambda$.

Cited by in corpus (1)

• Parallel non-linear iterations