Topic Archive: Laver functions
Say that a collection of Laver functions is jointly Laver if the functions can guess their targets simultaneously using just a single elementary embedding between them. In this talk we shall examine the notion of jointness in the simplest case of measurable cardinals, giving both equiconsistency results for the existence of large jointly Laver families and separating the existence of small such families from large ones. We shall also comment on how these results transfer to larger large cardinals, such as supercompact and strong cardinals, and, perhaps, how the notion of jointness may be interpreted for guessing principles not connected with large cardinals.
A joint diamond sequence on a cardinal $\kappa$ is a collection of $\diamondsuit_\kappa$ sequences which coheres in the sense that any collection of subsets of $\kappa$ may be guessed on stationary sets in some normal uniform filter on $\kappa$. This is the direct translation of joint Laver diamonds to smaller $\kappa$ which have no suitable elementary embeddings. In this talk I will show that, as opposed to the large cardinal case, joint diamond sequences simply exists whenever $\diamondsuit_\kappa$ holds.
Since Laver defined and used a Laver function to show that supercompact cardinals can be made indestructible by all $\lt\kappa$-directed closed forcing, Laver-like functions have been defined for various large cardinals and used for lifting embeddings in indestructibility arguments. Laver-like functions are also inherently interesting as guessing principles with affinity to $\diamondsuit$. Supposing that a large cardinal $\kappa$ can be characterized by the existence of some kind of embeddings, a Laver-like function $\ell:\kappa\to V_\kappa$ has, roughly speaking, the property that for any set $a$ in the universe, there is an embedding $j$ of the type characterizing the cardinal such that $j(\ell)(\kappa)=a$. Although Laver-like functions can be forced to exist for almost any large cardinal, only a few large cardinals including supercompact, strong, and extendible, have them outright. I will define the notion of a remarkable Laver function for a remarkable cardinal and show that every remarkable cardinal has a remarkable Laver function. Remarkable cardinals were introduced by Ralf Schindler who showed that a remarkable cardinal is precisely equiconsistent with the property that the theory of $L(\mathbb R)$ is absolute for proper forcing. Time permitting, I will show how the existence of remarkable Laver functions is used in demonstrating indestructibility for remarkable cardinals. This is joint work with Yong Cheng. An extended abstract can be found here.