The Gluing Property

TL;DR

The paper introduces the gluing property, a new compactness principle for measurable cardinals, establishing its relation to ultrafilters, extenders, and consistency strength, with precise results for specific cases.

Contribution

It defines the $ ext{gluing property}$ for measurable cardinals, relates it to ultrafilters and extenders, and determines the exact consistency strength for the $ ext{omega}$-gluing property.

Findings

Every $ ext{kappa}$-compact cardinal has the $2^ ext{kappa}$-gluing property.

Non-necessarily the $(2^ ext{kappa})^+$-gluing property.

The consistency strength for $ ext{kappa}$ to have the $ ext{omega}$-gluing property is $o( ext{kappa})= ext{omega}_1$.

Abstract

We introduce a new compactness principle which we call the gluing property. For a measurable cardinal $\kappa$ and a cardinal $\lambda$, we say that $\kappa$ has the $\lambda$-gluing property if every sequence of $\lambda$-many $\kappa$-complete ultrafilters on $\kappa$ can be glued into a $\kappa$-complete extender. We show that every $\kappa$-compact cardinal has the $2^\kappa$-gluing property, yet non-necessarily the $(2^\kappa)^+$-gluing property. Finally, we compute the exact consistency-strength for $\kappa$ to have the $\omega$-gluing property; this being $o(\kappa)=\omega_1$.