February 27
Brandon Ward,
CUNY
Failures of Choice in the Blurry HOD Hierarchy
Given a cardinal $\kappa$, a set is $\lt\kappa$-blurrily ordinal definable if it belongs to an OD set of cardinality less than $\kappa$, and the $\lt\kappa$-blurry HOD, denoted $\lt\kappa$-HOD, is the collection of all hereditarily $\lt\kappa$-blurrily OD sets. This is a weakly increasing hierarchy of inner models, beginning with HOD and whose union is the whole universe V (assuming choice in V). This hierarchy was introduced by Fuchs, with precursors (the cases $\kappa=\omega,\omega_1$) in the work of Hamkins and Leahy, and Tzouvaras. The leaps are the indices of the hierarchy where a new model occurs, and the possible structure of the leaps has been studied quite a bit by Fuchs, but the question whether the corresponding models satisfy the axiom of choice or not has not been investigated in generality so far. Let’s say that a leap is an AC-leap if the corresponding model in the blurry HOD hierarchy satisfies AC, and otherwise, it is a non-AC-leap. The main theme of this work is to gain a better understanding of the possible AC/non AC patterns in the structure of leaps.
Trivially, it is consistent that every level of the hierarchy satisfies choice, say, in a model of V=HOD (in which case there are no leaps). Meanwhile, it is part of the basic structure theory of leaps due to Fuchs that every limit of leaps is a non-AC-leap. It was observed by Hamkins and Leahy that (in the current terminology) $\lt\omega-HOD=HOD$, so $\omega$ is not a leap. The only published result on successor leaps which are non-AC leaps is due to Kanovei, whereby making use of a product of Jensen forcing a forcing extension of L is obtained in which $\omega_1$ is a non-AC leap. We will show two ways to generalize this construction to larger cardinals. The first obvious idea is to use the generalization of Jensen’s forcing to inaccessible $\kappa$ due to Friedman & Gitman in order to produce forcing extensions of $L$ where $\kappa^+$ is the least leap, and a non-AC leap, and GCH holds. The other generalization is to a cardinal of the form $\kappa^+$ such that $\kappa$ is regular and a certain $\diamondsuit$ assumption holds (which is always true in $L$ in this situation); the forcing is a free Suslin tree, and the argument that this works builds on recent work of Krueger.
Along the way, we will isolate the requisite properties of the forcings involved and arrive at the notions of $\kappa$-Kanovei and $\kappa$-Jensen posets.
This is joint work with my advisor, Gunter Fuchs.