Set Theory Seminar

A subset $X$ of a cardinal $\kappa$ is Ramsey if for every function $f:[X]^{<\omega}\rightarrow 2$ there is a set $H\subseteq X$ of cardinality $\kappa$ which is homogeneous for $f$, meaning that $f\upharpoonright[H]^n$ is constant for each $n<\omega$. Baumgartner proved that if $\kappa$ is Ramsey, then the collection of non-Ramsey subsets of $\kappa$ is a normal ideal on $\kappa$. We will discuss some recent results concerning Ramsey properties in which homogeneous sets are demanded to be indescribable of a particular degree. Moreover, by iterating Feng's Ramsey operator, which he used to define a notion of $\alpha$-Ramseyness of a cardinal $\kappa$, we will consider hypotheses in which homogeneous sets themselves satisfy various Ramsey properties. For ordinals $\alpha,\beta<\kappa$ we will define a notion of $\alpha$-$\Pi^1_\beta$-Ramseyness of a cardinal $\kappa$ where $\alpha$ indicates how many times the Ramsey operator has been iterated and $\beta$ indicates the degree of transfinite indescribability (due to Sharpe-Welch and independently Bagaria) one initially demands homogeneous to satisfy. We will prove that for $\alpha,\beta<\kappa$ an $\alpha$-$\Pi^1_\beta$-Ramsey cardinal is strictly between Feng's $\alpha$-Ramsey and an ($\alpha+1$)-Ramsey cardinal in consistency strength. Moreover, for fixed $\alpha<\kappa$, as $\beta$ increases the $\alpha$-$\Pi^1_\beta$-Ramsey cardinals yield a strictly increasing hierarchy, in a somewhat subtle sense. For $\beta_0<\beta_1<\kappa$ and for large enough $\alpha<\kappa$, $\kappa$ being $\alpha$-$\Pi^1_{\beta_0}$-Ramsey is equivalent to $\kappa$ being $\alpha$-$\Pi^1_{\beta_1}$-Ramsey (we will identify the least $\alpha$ at which this equivalence occurs). But if $\alpha,\beta_0<\kappa$ there is a large enough $\beta_1<\kappa$ such that $\kappa$ being $\alpha$-$\Pi^1_{\beta_0}$-Ramsey is strictly weaker than $\kappa$ being $\alpha$-$\Pi^1_{\beta_1}$-Ramsey. All of these results seem to require a careful analysis of the ideals associated to the various large cardinal notions.

We will present a method of constructing scales on products of regular cardinals. The choice of cardinals depends on the position of 'cutpoints' in the extender sequence of the core model. Here we will only discuss the simplest case of sequences of measures. Scale degrees will correspond to structures occurring in K that will be familiar to some from the construction of square sequences (both global and local) in fine structural inner models. The scale so constructed is canonical, short, and otherwise nice. This work is part of a joint project with Omer Ben-Neria.

In their 2004 paper, Moore, Hrusak and Dzamonja isolated a weakening of Jensen's diamond principle that could be 'parametrized' by a cardinal invariant, implies that the corresponding invariant is small, and yet is consistent with the failure of the Continuum Hypothesis. Moreover, these principles fully determine many cardinal invariants in 'canonical' models, those obtained by iterations of definable proper forcings. I will give a survey of this subject, and then describe a recent application in the study of maximal almost disjoint families of subspaces of a countably infinite-dimensional vector space.

In this talk I will discuss some of the definitions and results from the 1975 paper 'Flipping properties: A unifying thread in the theory of large cardinals' by Abramson, Harrington, Kleinberg, and Zwicker. I will introduce flipping properties, and show how they can be used to characterize inaccessible, weakly compact, and measurable cardinals.

This is joint work with John Clemens. We will begin this talk by discussing the problem of classifying the countable scattered linear orders. Here a linear order is called scattered if the rational order doesn’t embed into it. The class of such orders admits a ranking function valued in the ordinals; we will study the corresponding classification problem for each fixed rank. We will show that each increase in rank results in a “jump” in the complexity of the classification problem. In the second part of the talk we will define a family of jump operators on equivalence relations, each associated with a fixed countable group. The jump in the case of scattered linear orders is that associated with the group Z of integers. We will discuss the basic theory of these jump operators. Finally, we will discuss the question of when such a jump operator is proper, in the sense that the jump of E is strictly above E in the Borel reducibility order.

I will talk about a strengthening of a classical result of Foreman and Magidor, which states that any sequence of stationary subsets of distinct regular cardinals, each set consisting of ordinals of countable cofinality, is mutually stationary. The strengthening allows us to conclude a form of simultaneous reflection of stationarity which guarantees the existence of a mutually stationary sequence of exact reflection points, as a consequence of the subcomplete forcing axiom, and in fact of a weaker principle that corresponds to the subcomplete fragment of the well-known strong reflection principle.