Joint diamonds and Laver diamonds

Abstract

The concept of jointness for guessing principles, specifically ${♢}_{\kappa }$ and various Laver diamonds, is introduced. A family of guessing sequences is joint if the elements of any given sequence of targets may be simultaneously guessed by the members of the family. While equivalent in the case of ${♢}_{\kappa }$, joint Laver diamonds are nontrivial new objects. We give equiconsistency results for most of the large cardinals under consideration and prove sharp separations between joint Laver diamonds of different lengths in the case of $\theta$-supercompact cardinals.

34 pages; revised version with several improvements, including expanded Sections 3.3 and 4

References in corpus (1)

• Extensions with the approximation and cover properties have no new large cardinals