The Gluing Property

Abstract

We introduce a new compactness principle which we call the gluing property. For a measurable cardinal $\hat{I}º$ and a cardinal $\hat{I}»$, we say that $\hat{I}º$ has the $\hat{I}»$-gluing property if every sequence of $\hat{I}»$-many $\hat{I}º$-complete ultrafilters on $\hat{I}º$ can be glued into a $\hat{I}º$-complete extender. We show that every $\hat{I}º$-compact cardinal has the $2^{\hat{I}}º$-gluing property, yet non-necessarily the $(2^{\hat{I}}º{)}^{+}$-gluing property. Finally, we compute the exact consistency-strength for $\hat{I}º$ to have the $\ddot{I}‰$-gluing property; this being $o(\hat{I}º)=\ddot{I}{‰}_1$.

Lemma 3.5 of the previous version was not correct. We have removed it and tweaked the section accordingly