UCLA

Mutual and tight stationarity are properties akin to the usual notion of stationarity, but defined for sequences of subsets of different regular cardinals. This work focuses particularly on tight stationarity, providing a new characterization for it and comparing it to other concepts of stationarity.

Starting from a pcf-theoretic scale, we define a transfer function mapping sequences of subsets to a single subset of a certain regular cardinal, the length of the scale. The transfer function preserves stationarity, in the sense that a sequence is tightly stationary if and only if it is mapped to a stationary subset.

Using this characterization, we explore the question of whether it is consistent that there exists a sequence of cardinals for which every stationary sequence (i.e., a sequence of subsets, each of which is stationary in the corresponding cardinal) is tightly stationary, and prove some results which give a negative answer in certain cases. We prove that adding Cohen reals introduces stationary sequences which are not tightly stationary, and in the extension by adding uncountably many Cohen reals, every sequence of cardinals has a stationary but not tightly stationary sequence. From a tree-like scale we construct a sequence of stationary sets that is not tightly stationary in a strong way, namely, its image under the transfer function is empty.

Investigating this question in the Prikry model, we define the notion of a forgetful sequence and prove that every forgetful sequence of cardinals has a stationary, not tightly stationary sequence. Along the way, we will analyze the scales which appear in the Prikry model.

Then we consider the question of Cummings, Foreman, and Magidor of whether it is consistent that there is a sequence of cardinals on which every mutually stationary sequence is tightly stationary. We prove that it is consistent that there is no such sequence of cardinals. This uses a supercompact version of a construction adapted from Koepke which ensures that every stationary sequence is mutually stationary, provided that there is enough space between successive cardinals of the underlying sequence. Furthermore, this property of the model is indestructible under further Prikry forcing, which suggests that it is difficult to obtain a positive answer to the CFM question. The results in this section were obtained jointly with Itay Neeman.

Finally, we explore the combinatorics of tight stationarity. This leads to the notion of a careful set, which is a strengthening of being in the range of the transfer function. We produce a model where there is a singular cardinal for which all subsets of the successor are careful, which suffices to prove a splitting result for tightly stationary sequences. Using a version of the diagonal supercompact Prikry forcing, we obtain such a model where the singular cardinal is strong limit. These results start from a model with a continuous tree-like scale on the singular cardinal.