Set Theory Seminar

Prikry forcing was devised by Karel Prikry to show that, given large cardinals, one can make a regular cardinal singular while having it remain a cardinal. The cofinal sequence added by Prikry forcing is called the Prikry sequence. The Prikry sequence is maximal in the sense that any other generic sequence is, modulo a finite initial segment, a subsequence of the Prikry sequence. We call this property the maximality property. A key to showing this property is using the normal ultrafilter associated with the singularized large cardinal. In this talk, we discuss the maximality property for Prikry forcings of various ultrafilters. We present partial results on a conjecture of Woodin on maximality for supercompact Prikry forcing, Prikry forcings without the maximality property, and intermediate models of Prikry forcings.

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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.

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Canjar and Laflamme studied the equivalent notions of Canjar ultrafilters and Strong P-points (resp.). These are ultrafilters with the property that the respective Mathias forcing does not add a dominating real. They showed that such ultrafilters must be P-points without rapid RK-predecessors, and conjectured that these two properties in fact characterize Canjar ultrafilters. While the Canjar-Laflamme conjecture was proven to be false by Blass-Hrusak-Verner, we present here a characterization of Canjar ultrafilters that catches the underlying intuition of the Canjar-Laflamme conjecture using the cofinal type of $\omega^{\omega}$ with the everywhere domination order. After proving this theorem, we will present several applications of our characterization, including a classification of the class of Tukey-idempotent ultrafilters and an answer to a question of Hrusak-Verner about the possibility of $P(\omega)/I$ adding a Canjar ultrafilter, where $I$ is analytic. This is joint work with Natasha Dobrinen and Tan Ozlap.

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We study first-order reflection principles in ZF set theory with urelements (ZFU). The Reflection Principle (RP) asserts that every formula is absolute between the universe and some transitive set, while the Partial Reflection Principle (RP-) states that every true assertion is true in some transitive set. It is known that RP-, RP, and the Collection Principle are equivalent over ZFU with the Axiom of Choice, yet none of these principles is provable. We separate these three principles in ZFU: Collection and RP- are independent of one another, and Collection together with RP- does not imply RP. We also show that RP and Collection are equivalent assuming either the Tail axiom or Small Violations of Choice. This is joint work with Elliot Glazer.

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Since its inception, set theory was concerned with the generalisation of the arithmetic operations on the natural numbers to the transfinite. There are two different such generalisations, though, the ordinal and cardinal one. Unlike in the case of the latter, the literature on the former is somewhat sparse. Also, very roughly, more is known about ordinals than general linear order types, more is known about the behaviour of addition than that of multiplication, and more is known about equality than embeddability. To illustrate this, we are introducing multiplicative analogues of indecomposability for linear order types that we call untranscendability and $s$-untranscendability, respectively. With the unique exception of the two-point type, every untranscendable type is indecomposable, and every $\sigma$-scattered untranscendable type is strongly indecomposable. Moreover, according to the Proper Forcing Axiom, every untranscendable Aronszajn type is strongly indecomposable. Finally, a theorem of Hagendorf and Jullien to the effect that every strictly indecomposable type must be strictly indecomposable to either the left or right turns out to have a natural multiplicative analogue for $s$-untranscendable types. I am going to describe all our results in context and sketch possibilities for future research.

This is joint work with Garrett Ervin and Alberto Marcone.

Slides

I will discuss certain systems of models of two different types satisfying natural structural requirements. I will also present some recent uses of such systems as side conditions in forcing constructions, in (separate) joint work with Curial Gallart and with Ralf Schindler.

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We show that under a strong version of Laver generic large cardinal axiom, there is a model of multiverse theory of Steel with the initial universe being an elementary submodel of the bedrock, and such that the model of multiverse contains (practically) all forcable situations over the (real) universe.

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