Some remarks on tall cardinals, indestructibility, and equiconsistency

Abstract

The ultimate goal of this note is to establish results pointing to our concluding conjecture that instances of tallness are equiconsistent with certain failures of $\mathsf{GCH}$ at a measurable cardinal. Towards that end, we begin by showing that any tall cardinal can have its tallness made indestructible under Sacks forcing, and that the construction used can be iterated so as to produce a model containing a (possibly proper) class of tall cardinals in which each member of the class has its tallness indestructible under Sacks forcing. We then make precise Hamkins' proof sketch given in Corollary 3.14 of "Tall cardinals" (2009) that the theories $\mathsf{ZFC} + {}$"There is a tall cardinal" and $\mathsf{ZFC} + {}$"There is a strong cardinal" are equiconsistent. We finish by proving two theorems concerning equiconsistency, instances of tallness, and failures of $\mathsf{GCH}$ that provide the basis for our concluding conjecture.