Controlling cardinal characteristics without adding reals

TL;DR

This paper explores how certain cardinal characteristics of the continuum can be independently manipulated through extensions that do not add new small sequences, demonstrating the possibility of thirteen distinct characteristic values.

Contribution

It shows the consistency of separating multiple cardinal characteristics simultaneously without adding reals, including the full separation in Cichoń's diagram and other MA-number distinctions.

Findings

Thirteen different values for cardinal characteristics are consistent.

Separation of MA for k-Knaster from MA for (k+1)-Knaster.

Separation of MA for union of all k-Knaster forcings from MA for precaliber.

Abstract

We investigate the behavior of cardinal characteristics of the reals under extensions that do not add new ${<}\kappa$-sequences (for some regular $\kappa$). As an application, we show that consistently the following cardinal characteristics can be different: The ("independent") characteristics in Cicho\'n's diagram, plus $\aleph_1<\mathfrak m<\mathfrak p<\mathfrak h<\mathrm{add}(\mathcal{N})$. (So we get thirteen different values, including $\aleph_1$ and continuum). We also give constructions to alternatively separate other MA-numbers (instead of $\mathfrak m$), namely: MA for $k$-Knaster from MA for $k+1$-Knaster; and MA for the union of all $k$-Knaster forcings from MA for precaliber.