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.

Coding information into the structure of the universe is a forcing technique with many applications in set theory. To carry out it out, we a need a property that: i) can be easily switched on or off at (e.g.) each regular cardinal in turn, and ii) is robust with regards both to small and to highly-closed forcing. GCH coding, controlling the success or failure of the GCH at each cardinal in turn, is the most widely used, and for good reason: there are simple forcings that turn it on and off, and it is easily seen to be unaffected by small or highly-closed forcing. However, it does have limitations – most obviously, GCH coding is of necessity incompatible with the GCH itself. In this talk I will present an alternative coding using the property Diamond*, a variant of the classic Diamond. I will discuss Diamond* and demonstrate that it satisfies the requirements for coding while preserving the GCH.

Although the basic techniques for controlling Diamond* have been known for some time, to my knowledge the first use of Diamond* as a coding axiom was by Andrew Brooke-Taylor in his work on definable well-orders of the universe. I will follow the excellent exposition presented in his dissertation.

A Laver diamond for a given large cardinal $\kappa$ is a function $\ell$, defined on $\kappa$, such that $j(\ell)(\kappa)$ can take any reasonable value, where $j$ is a relevant large cardinal embedding. A sequence of such functions is called jointly Laver or a joint Laver diamond if they can be made to take any given sequence of such values at the same time via a single embedding. In the talk we will consider questions about when such sequences outright exist, when their existence is equiconsistent with and when their existence is consistency-wise strictly stronger than the large cardinal in question.

The speaker will give the second part of her talk, continued from the previous week. An $\omega_1$-like model of set theory is an uncountable model, all of whose initial segments are countable. The speaker will present two $\omega_1$-like models of set theory, constructed using $\Diamond$, which are incomparable with respect to embeddability: neither is isomorphic to a submodel of the other. Under a suitable large cardinal assumption, there are such models that are well-founded.

An $\omega_1$-like model of set theory is an uncountable model, all of whose initial segments are countable. The speaker will present two $\omega_1$-like models of set theory, constructed using $\Diamond$, which are incomparable with respect to embeddability: neither is isomorphic to a submodel of the other. Under a suitable large cardinal assumption, there are such models that are well-founded.

I will show that Diamond Plus holds in inner models of the form L[A], for subsets A of aleph one in the sense of L[A]. Putting this together with the result from last meeting, that Diamond Plus implies the Kurepa Hypothesis, I will show that if the Kurepa Hypothesis fails, then aleph two is an inaccessible cardinal in L. Again, putting this together with another result from the previous seminar meeting, that one can force the failure of Kurepa’s Hypothesis over a model with an inaccessible cardinal, this shows the equiconsistency of the failure of Kurepa’s Hypothesis with an inaccessible cardinal, over ZFC. These results are mainly due to Silver and Solovay.

The speaker will prove some connections between Diamond Plus and the Kurepa Hypothesis.