# Seminars (HIFW01)

Videos and presentation materials from other INI events are also available.

Search seminar archive

Event When Speaker Title Presentation Material
HIFW01 24th August 2015
10:00 to 11:00
101 Years of Modern Set Theory: Felix Hausdorff's "Foundations of Set Theory"
Felix Hausdorff's 1914 monograph "Grundzüge der Mengenlehre" (Foundations of Set Theory) marks the beginning of modern set theory:

- it is the first comprehensive presentation of set theory as a mature and rich mathematical field, untroubled by philosophical worries about infinity and paradoxes;

- it introduces fundamental notions and theories that have been crucial for the further development of set theory and its applications, notably the axioms for topological spaces;

- even today, a considerable part of research in set theory has direct links to Hausdorff's work.

In my talk I shall describe Hausdorff's book and locate it in the history of set theory. I shall also present his views on the foundations of mathematics and set theory. HIFW01 24th August 2015
11:30 to 12:30
The tree property (session 1)
The tree propperty at $\kappa$ says that every tree of height $\kappa$ and levels of size less than $\kappa$ has a cofinal branch. A long term project in set theory is to get the consistency of the tree property at every regular cardinal greater than $\aleph_1$. So far we only know that it is possible to have the tree property up to $\aleph_{\omega+1}$, due to Neeman. The next big hurdle is to obtain it both at $\aleph_{\omega+1}$ and $\aleph_{omega+2}$ when $\aleph_\omega$ is trong limit. Doing so would require violating the singular cardinal hypothesis at $\aleph_omega$. In this tutorial we will start with some classic facts about the tree property, focusing on branch lemmas, successors of singulars and Prikry type forcing used to negate SCH. We will then go over recent developments including a dichotomy theorem about which forcing posets are good candidates for getting the tree property at $\aleph_{\omega+1}$ together with not SCH at $\aleph_\omega$. Finally, we will discuss the problem of obtaining the tree property at the first and double successors of a singular cardinal simultaneously. HIFW01 24th August 2015
13:30 to 14:00
What is a forcing extension (of V)?
Recent research into the representability of forcing extensions within ground models has often focussed on semantic formulations; we find definable class models within V that express very closely what it means to be a forcing extension of V (such as Hamkins' Naturalist Account of Forcing). In this paper, I argue that while this formulation appears to be a good candidate for interpreting forcing over V, it is problematic in that it is unable to interpret theorems concerning objects below a measurable cardinal whilst keeping the ultrapower well-founded, and also encounters difficulties in interpreting class forcings. Instead I suggest that the motivation of a strong class theory (such as MK) over V, combined with a syntactic approach to forcing in a strengthened logic, is able to provide an interpretation of various forcing constructions which does not encounter these difficulties. HIFW01 24th August 2015
13:30 to 14:00
A generalisation of closed unbounded sets
A generalisation of stationarity, associated with stationary reflection, was introduced in . I give an alternative characterisation of these $n$-stationary sets by defining a generalisation of closed unbounded (club) sets, so an $n$-stationary set is defined in terms of these $n$-clubs in the usual way. I will then look into what familiar properties of stationary and club sets will still hold in this more general setting, and explore the connection between these concepts and indescribable cardinals. Many of the simpler properties generalise completely, but for others we need an extra assumption. For instance to generalise the splitting property of stationary sets we have: If $\ kappa$ is $\ Pi^1$ $n$$-1 indescribable, then any n-stationary subset of \kappa is the union of \ kappa many pairwise-disjoint n-stationary sets. In L these properties generalise straightforwardly as there any cardinal which admits an n-stationary set is \ Pi^1_{n-1} indescribable  . If there is time I will also introduce a generalisation of ineffable cardinals and a weak \diamond principal that is associated.  J. Bagaria, M. Magidor, and H. Sakai. Reflection and indescribability in the constructible universe. \textit{Israel Journal of Mathematics}, to appear (2012). HIFW01 24th August 2015 14:00 to 14:30 A Blaszczyk Topological representation of lattice homomorphisms Wallman proved that if \mathbb{L} is a distributive lattice with \mathbf{0} and \mathbf{1}, then there is a T_1-space with a base (for closed subsets) being a homomorphic image of \mathbb{L}. We show that this theorem can be extended over homomorphisms. More precisely: if \bf{Lat} denotes the category of normal and distributive lattices with \mathbf{0} and \mathbf{1} and homomorphisms, and \bf{Comp} denotes the category of compact Hausdorff spaces and continuous mappings, then there exists a contravariant functor \mathcal{W}:\bf{Lat}\to\bf{Comp}. When restricted to the subcategory of Boolean lattices this functor coincides with a well-known Stone functor which realizes the Stone Duality. The functor \mathcal{W} carries monomorphisms into surjections. However, it does not carry epimorphisms into injections. The last property makes a difference with the Stone functor. Some applications to topological constructions are given as well. HIFW01 24th August 2015 14:00 to 14:30 Hyperclass Forcing in Morse Kelley Set Theory There are mainly two different types: set-forcing and class-forcing, where the forcing notion is a set or class respectively. Here, we want to introduce and study the next step in this classification by size, namely hyperclass-forcing (where the conditions of the forcing notion are themselves classes) in the context of an extension of Morse-Kelley class theory, called MK^*. We define this forcing by using a symmetry between MK^* models and models of ZFC^- plus there exists a strongly inaccessible cardinal (called SetMK^*). We develop a coding between \beta-models \mathcal{M} of MK^* and transitive models M^+ of SetMK^* which will allow us to go from \mathcal{M} to M^+ and vice versa. So instead of forcing with a hyperclass in MK^* we can force over the corresponding SetMK^* model with a class of conditions. For class-forcing to work in the context of ZFC^- we show that the SetMK^* model M^+ can be forced to look like L_{\kappa^*}[X], where \kappa^* is the height of M^+, \kappa strongly inaccessible in M^+ and X\subseteq\kappa. Over such a model we can apply class-forcing and we arrive at an extension of M^+ from which we can go back to the corresponding \beta-model of MK^*, which will in turn be an extension of the original \mathcal{M}. We conclude by giving an application of this forcing in sho wing that every \beta-model of MK^* can be extended to a minimal \beta-model of MK^* with the same ordinals. HIFW01 24th August 2015 15:00 to 16:00 A D Törnquist Around the definability of mad families I will talk about my new proof that there are no analytic infinite mad (maximal almost disjoint) families of subsets of \omega, a result originally proved by A.D.R. Mathias in his famous "Happy Families" paper. The new proof motivates a proof that there are no infinite mad families in Solovay's model. If time permits, I will also talk about the differences between mad families of subsets of \omega and other types of mad families, such as eventually different families of functions from \omega to \omega, and why these problems may be very different from the situation presented by mad families of subsets of \omega. HIFW01 24th August 2015 16:00 to 17:00 The Hurewicz dichotomy for generalized Baire spaces By classical results of Hurewicz, Kechris and Saint-Raymond, an analytic subset of a Polish space X is covered by a Ksigma subset of X if and only if it does not contain a closed-in-X subset homeomorphic to the Baire space omega^omega. We consider the analogous statement (which we call Hurewicz dichotomy) for Sigma11 subsets of the generalized Baire space kappa^kappa for a given uncountable cardinal kappa with kappa=kappa^(<kappa), and show how to force it to be true in a cardinal and cofinality preserving extension of the ground model. Moreover, we show that if the GCH holds, then there is a cardinal preserving class forcing extension in which the Hurewicz dichotomy for Sigma11 subsets of kappa^kappa holds at all uncountable regular cardinals kappa, while strongly unfoldable and supercompact cardinals are preserved. On the other hand, in the constructible universe L the dichotomy for Sigma11 sets fails at all uncountable regular cardinals, and the same happens in any generic extension obtained by adding a Cohen real to a model of GCH. This is joint work with Philipp Lücke and Luca Motto Ros. HIFW01 25th August 2015 09:00 to 10:00 P Koellner The Search for Deep Inconsistency The hierarchy of large cardinals provides us with a canonical means to climb the hierarchy of consistency strength. There have been any purported inconsistency proofs of various large cardinal axioms. For example, there have been many proofs purporting to show that measurable cardinals are inconsistent. But to date the only proofs that have stood the test of time are those which are rather transparent and simple, the most notable example being Kunen's proof showing that Reinhardt cardinals are inconsistent. The Kunen result, however, makes use of AC. And long standing open question is whether Reinhardt cardinals are consistent in the context of ZF. In this talk I will survey the simple inconsistency proofs and then raise the question of whether perhaps the large cardinal hierarchy outstrips AC, passing through Reinhardt cardinals and reaching far beyond. There are two main motivations for this investigation. First, it is of interest in its own right to determine whether the hierarchy of consistency strength outstrips AC. Perhaps there is an entire "choicless" large cardinal hierarchy, one which reaches new consistency strengths and has fruitful applications. Second, since the task of proving an inconsistency result becomes easier as one strengthens the hypothesis, in the search for a deep inconsistency it is reasonable to start with outlandishly strong large cardinal assumptions and then work ones way down. This will lead to the formulation of large cardinal axioms (in the context of ZF) that start at the level of a Reinhardt cardinal and pass upward through Berkeley cardinals (due to Woodin) and far beyond. Bagaria, Woodin, and myself have been charting out this new hierarchy. I will discuss what we have found so far. HIFW01 25th August 2015 10:00 to 11:00 Local Ramsey Theory in abstract spaces We study the notion of semiselectiviy for coideals of the space of infinite sets of natural numbers, and propose a generalization to abstract Ramsey spaces. We also consider the corresponding forcing notions. Abstract Ramsey spaces in the sense of Todorcevic  provide a general framework that unifies several Ramsey type results.  Todorcevic, S. Introduction to Ramsey spaces. Princeton University Press, 2010. HIFW01 25th August 2015 11:30 to 12:30 The tree property (session 2) The tree propperty at \kappa says that every tree of height \kappa and levels of size less than \kappa has a cofinal branch. A long term project in set theory is to get the consistency of the tree property at every regular cardinal greater than \aleph_1. So far we only know that it is possible to have the tree property up to \aleph_{\omega+1}, due to Neeman. The next big hurdle is to obtain it both at \aleph_{\omega+1} and \aleph_{omega+2} when \aleph_\omega is trong limit. Doing so would require violating the singular cardinal hypothesis at \aleph_omega. In this tutorial we will start with some classic facts about the tree property, focusing on branch lemmas, successors of singulars and Prikry type forcing used to negate SCH. We will then go over recent developments including a dichotomy theorem about which forcing posets are good candidates for getting the tree property at \aleph_{\omega+1} together with not SCH at \aleph_\omega. Finally, we will discuss the problem of obtaining the tree property at the first and double successors of a singular cardinal simultaneously. HIFW01 25th August 2015 13:30 to 14:00 Generic I0 at \aleph_\omega It is common practice to consider the generic version of large cardinals defined with an elementary embedding, but what happens when such cardinals are really large? The talk will concern a form of generic I0 and the consequences of this extravagant hypothesis on the "largeness" of the powerset of \aleph_\omega. This research is a result of discussions with Hugh Woodin. HIFW01 25th August 2015 14:00 to 14:30 Forcing, regularity properties and the axiom of choice We consider general regularity properties associated with Suslin ccc forcing notions. By Solovay's celebrated work, starting from a model of ZFC+"There exists an inaccessible cardinal", we can get a model of ZF+DC+"All sets of reals are Lebesgue measurable and have the Baire property". By another famous result of Shelah, ZF+DC+"All sets of reals have the Baire property" is equiconsistent with ZFC. This result was obtained by isolating the notion of "sweetness", a strong version of ccc which is preserved under amalgamation, thus allowing the construction of a suitably homogeneous forcing notion. The above results lead to the following question: Can we get a similar result for non-sweet ccc forcing notions without using an inaccessible cardinal? In our work we give a positive answer by constructing a suitable ccc creature forcing and iterating along a non-wellfounded homogeneous linear order. While the resulting model satisfies ZF+\neg AC_{\omega}, we prove in a subsequent work that starting with a model of ZFC+"There is a measurable cardinal", we can get a model of ZF+DC_{\omega_1}. This is joint work with Saharon Shelah. HIFW01 25th August 2015 14:00 to 14:30 S Friedman Forcing failures of covering in HOD Inspired by questions about the HOD dichotomy, we consider how and in what manner we can force HOD to be "far from V". Our initial result is an equiconsistency in ZFC between a model with a proper class of measurable cardinals and HOD and V agreeing on the cardinals yet disagreeing on the cofinality of a proper class of cardinals. HIFW01 25th August 2015 15:00 to 16:00 Symmetries Abstract: In the last years there has been a second boom of the technique of forcing with side conditions (see for instance the recent works of Asperó-Mota, Krueger and Neeman describing three different perspectives of this technique). The first boom took place in the 1980s when Todorcevic discovered a method of forcing in which elementary substructures are included in the conditions of a forcing poset to ensure that the forcing poset preserves cardinals. More than twenty years later, Friedman and Mitchell independently took the first step in generalizing the method from adding small (of size at most the first uncountable cardinal) generic objects to adding larger objects by defining forcing posets with finite conditions for adding a club subset on the second uncountable cardinal. However, neither of these results show how to force (with side conditions together with another finite set of objects) the existence of such a large object together with the continuum being small. In the first part of this talk I will discuss new results in this area. This is joint work with John Krueger improving the symmetric CH preservation argument previously made by Asperó and Mota. In the second part of this talk I will use generalized symmetric systems in order to prove that, for each regular cardinal k, there is a poset P_k forcing the existence of a (k,k++)-superatomic boolean algebra. This is joint work with William Weiss inspired in an unpublished note from September 2009 where Asperó and Bagaria introduced the forcing P_{\omega}. HIFW01 25th August 2015 16:00 to 17:00 P Holy Failures of the Forcing Theorem The forcing theorem is the most fundamental result about set forcing, stating that the forcing relation for any set forcing is definable and that the truth lemma holds, that is everything that holds in a generic extension is forced by a condition in the relevant generic filter. We show that both the definability of the forcing relation and the truth lemma can fail for class forcing. We will also present positive results about and characterizations of the forcing theorem in class forcing. This is joint work with Regula Krapf, Philipp Lücke, Ana Njegomir and Philipp Schlicht. HIFW01 26th August 2015 09:00 to 10:00 Hausdorff Medal Award HIFW01 26th August 2015 10:00 to 11:00 Hausdorff Medal Lecture HIFW01 26th August 2015 11:30 to 12:30 The tree property (session 3) The tree propperty at \kappa says that every tree of height \kappa and levels of size less than \kappa has a cofinal branch. A long term project in set theory is to get the consistency of the tree property at every regular cardinal greater than \aleph_1. So far we only know that it is possible to have the tree property up to \aleph_{\omega+1}, due to Neeman. The next big hurdle is to obtain it both at \aleph_{\omega+1} and \aleph_{\omega+2} when \aleph_\omega is trong limit. Doing so would require violating the singular cardinal hypothesis at \aleph_\omega. In this tutorial we will start with some classic facts about the tree property, focusing on branch lemmas, successors of singulars and Prikry type forcing used to negate SCH. We will then go over recent developments including a dichotomy theorem about which forcing posets are good candidates for getting the tree property at \aleph_{\omega+1} together with not SCH at \aleph_\omega. Finally, we will discuss the problem of obtaining the tree property at the first and double successors of a singular cardinal simultaneously. HIFW01 26th August 2015 13:30 to 14:00 D Ikegami Universally Baire subsets of 2^{\kappa} In this talk, we present a basic theory of universally Baire subsets of 2^{\kappa} which generalize that of universally Baire sets of reals due to Feng, Magidor, and Woodin. This is joint work with Matteo Viale. HIFW01 26th August 2015 13:30 to 14:00 Cofinalities of Marczewski Ideals We study the cofinality number of sigma-ideals related to the Marczewski-ideal on the Baire and Cantor space. These ideals are not Borel-generated, and the cofinality number is usually greater than the continuum. This is joint work with Jörg Brendle and Wolfgang Wohofsky. HIFW01 26th August 2015 14:00 to 14:30 Partition Relation Equiconsistent with \exists \kappa(o(\kappa) = \kappa^+) Preamble: In this work we deal with partition relations with infinite exponents under ZFC, hence all results are limited to definable functions. In , M. Spector has proven, basically, that \exists \kappa(o(\kappa) = 1) is equiconsistent with \aleph_1 \rightarrow (\omega)^{\omega}_{\aleph_0}. In , we were able to show that the result generalizes to n = 2; namely, \exists \kappa(o(\kappa) = 2) is equiconsistent with \aleph_1 \rightarrow (\omega^2)^{\omega^2}_{\aleph_0}. Surprisingly at first sight, this property cannot be generalized further (for n> 2), and later on we were able to prove that \aleph_1 \rightarrow (\omega^3)^{\omega^3}_{\aleph_0} is equiconsistent with \exists \kappa(o(o(\kappa)) = 2). The above lead us to a finer notion of homogeneity: Definition: Weak Homogeneity is the partition property \kappa \xrightarrow{\text{\tiny WH}}(\lambda)^{\eta}_{\mu} where the only considered subsequences of \lambda are those that are created by removing (or, complementarily, collecting) only finitely many segments of \lambda. Using week homogeneity we were able to prove the following for any ordinal \alpha : \exists \kappa(o(\kappa) = \alpha) is equiconsistent with \aleph_1 \xrightarrow{\text{\tiny WH}}(\omega^\alpha)^{\omega^\alpha}_{\aleph_0}. Later on we were able to characterize the consistency strength of \exists \kappa(o(\kappa)~=~\kappa), and recently we have arrived at the main result of this paper: \exists \kappa(o(\kappa) = \kappa^+) is equiconsistent with \aleph_1 \xrightarrow{\text{\tiny WH}}(\aleph_1)^{\aleph_1}_{\aleph_0} References:  M. Spector: Natural Sentences of Mathematics which are independent of V = L, V = L^\mu etc., 1978 (preprint).  Y.M. Kimchi: Dissertation, 1987, The Hebrew University of Jerusalem, Israel HIFW01 26th August 2015 14:00 to 14:30 On the class of perfectly null sets and its transitive version The ideals of universally null sets (UN, sets which are null with respect to any Borel diffused measure) and perfectly meager sets (PM, sets which are meager when restricted to any perfect set) are best known among the classes of special subsets of the real line. Those two ideals were long considered to be somehow dual, though some differences were also known. P. Zakrzewski proved that two other earlier defined classes of sets smaller then PM coincide and are dual to UN. Therefore he proposed to call this class universally meager sets. The PM class was left without a counterpart, and we try to define a class of sets which may play the role of a dual class to PM and we also consider its transitive version. I will present some properties of those classes and give few important problems which are still open. HIFW01 27th August 2015 09:00 to 10:00 Mathias and Set Theory On Mathias Day, the life work of Adrian Mathias in set theory will be surveyed and celebrated, in full range and extent.  HIFW01 27th August 2015 10:00 to 11:00 C Delhomme The relation of attack When he visited Barcelona, Adrian Mathias got interested in questions regarding the relation of attack considered by dynamicists. Given a self-mapping of a topological space, a point x attacks y, or y is an omega-limit point of x, if y is a cluster point of the sequence of iterates of x. He successfully applied set-theoretical ideas in the study of this relation. We shall describe some of his contributions to the subject. HIFW01 27th August 2015 11:30 to 12:30 Happy Families and Their Relatives We first recall the notion of \ happy$$\ families$ as well as their combinatorial properties. Then we present some families which are related to $\ happy$$\ families and investigate \ Mathias$$\ forcing$ restricted to these families. In the second part we show the relation between $\it Mathias$$\ forcing and the \ Ramsey$$\ property$ and discuss the still open problem whether one can take $\ Mathias'$$\ inaccessible away. In the last part, we sketch Shelah's construction of a model of ZFC in which there are exactly \ 70$$\ happy$$\ ultrafilters$. HIFW01 27th August 2015
13:30 to 14:00
Selective properties of ideals
We will discuss several selective properties of ideals on countable sets. In the case of maximal ideals all those properties coincide and are equivalent to the notion of selectiveness of a maximal ideal (a maximal ideal is selective iff its dual filter is a Ramsey ultrafilter), however in general those selective properties differ from each other. We will show some of their connections to ideal convergence of sequences of functions, descriptive complexity of ideals and topological ways of representing ideals. HIFW01 27th August 2015
13:30 to 14:00
Connecting topological dimension theory and recursion theory
We introduce the point degree spectrum of a represented spaces as a substructure of the Medvedev degrees, which integrates the notion of Turing degrees, enumeration degrees, continuous degrees, and so on. The point degree spectrum connects descriptive set theory, topological dimension theory and computability theory. Through this new connection, for instance, we construct a family of continuum many infinite dimensional Cantor manifolds possessing Haver's property C whose Borel structures at an arbitrary finite rank are mutually non-isomorphic, which strengthen various theorems in infinite dimensional topology such as Roman Pol's solution to Pavel Alexandrov's old problem. HIFW01 27th August 2015
14:00 to 14:30
Chain conditions, layered partial orders and weak compactness
Motivated by a conjecture of Todorcevic, we study strengthenings of the $\kappa-chain$ conditions that are equivalent to the $\kappa-chain$ condition in the case where $\kappa$ is a weakly compact cardinal. We then use such properties to provide new characterizations of weakly compact cardinals. In addition, we show that the question whether weak compactness is characterized by the statement that all $\kappa-Knaster$ posets satisfy these properties is independent from the axioms of ZFC. This is joint work with Sean D. Cox (VCU Richmond). HIFW01 27th August 2015
15:00 to 16:00
G Fuchs Prikry type sequences: a composition of interconnected results
I would like to survey a series of beautiful and almost mysterious properties of Prikry sequences which have analogues for other Prikry type forcings. The first of these is Mathias' characterization of Prikry sequences as those that are almost contained in every set of measure 1 with respect to the normal ultrafilter being used for the forcing. This is the key to the second property, which is that the sequence of critical points when forming iterated ultrapowers by that ultrafilter is a Prikry sequence over the limit model. Using this, it is not hard to conclude that Prikry sequences are maximal, in the sense that they almost contain every other Prikry sequence present in their forcing extension. Another phenomenon is that the forcing extension of the limit model by the critical sequence is the same as the intersection of the finite iterates. I will show another canonical representation of that model. Yet another property is that the limit model can be realized as a single Boolean ultrapower. Most of these results were known for Prikry forcing, and I will show that some of them carry over to certain variants of Prikry forcing and Magidor forcing. HIFW01 27th August 2015
16:00 to 17:00
A super-Dowker filter
A super-Dowker filter is a filter F on a set X such that

1) For every sequence of F-large sets there are x,y distinct with x in A_y and y in A_x

2) For every partition of X into two parts there exist a sequence as in 1) and a cell of the partition such that all pairs as in 1) lie in this cell

Building on work of Balogh and Gruenhage we show the consistency of the existence of a super-Dowker filter HIFW01 28th August 2015
09:00 to 10:00
Infinite dimensional Ellentuck spaces

Topological Ramsey spaces have proved essential to solving certain problems in Banach spaces, Graph Theory, Set Theory, and Topology. In recent years, they have provided a mechanism for investigating initial Rudin-Keisler and Tukey structures in the Stone-Cech compactification of the natural numbers. In work of Dobrinen and Todorcevic and work of Dobrinen, Mijares, Trujillo, certain partial orders which force ultrafilters with partition relations were found to be equivalent to some new classes of topological Ramsey spaces. These in turn were used to precisely investigate the Ramsey-theoretic properties and the Tukey and Rudin-Keisler structures of the associated ultrafilters.

The Ellentuck space is the quintessential example of a topological Ramsey space. It is closely connected with Mathias forcing and with $\mathcal{P}(\omega)/\mathrm{Fin}$ which forces a Ramsey ultrafilter. Building on work in [Blass/Dobrinen/Raghavan15] investigating the Tukey type of the ultrafilter forced by $\mathcal{P}(\omega\times\omega)/\mathrm{Fin}\otimes\mathrm{Fin}$, we found in [DobrinenJSL15] that the essential structure responsible for the forcing properties actually is a 2-dimensional version of the Ellentuck space.

In this talk, we will present work in [DobrinenIDE15] constructing a new class of topological Ramsey spaces which may be viewed as infinite dimensional Ellentuck spaces. We will then present the Ramsey-classification theorems for equivalence relations on fronts and some applications to their related $\sigma$-closed forcings and the Rudin-Keisler and Tukey structures below the generic ultrafilters.

• [Blass/Dobrinen/Raghavan15] Andreas Blass, Natasha Dobrinen, and Dilip Raghavan, The next best thing to a p-point, Journal of Symbolic Logic 80, no.3, 866-900.
• [DobrinenJSL15] Natasha Dobrinen, High dimensional Ellentuck spaces and initial chains in the Tukey structure of non-p-points, Journal of Symbolic Logic, 27pp, To appear.
• [DobrinenIDE15] Natasha Dobrinen, Infinite dimensional Ellentuck spaces, 35 + pp, Preprint. HIFW01 28th August 2015
10:00 to 11:00
M Goldstern Ultrafilters without p-point quotients
A p-point is a nonprincipal ultrafilter on the set N of natural numbers which has the property that for every countable family of filter sets there is a pseudointersection in the filter, i.e. a filter set which is almost contained in each set of the family. Equivalently, a p-point is an element of the Stone-Cech remainder beta(N) minus N whose neighborhood filter is closed under countable intersections.

It is well known that p-points "survive" various forcing iterations, that is: extending a universe V with certain forcing iterations P will result in a universe V' in which all (or at least: certain well-chosen) p-points are still ultrafilter bases in the extension. This shows that the sentence "The continuum hypothesis is false, yet there are aleph1-generated ultrafilters, namely: certain p-points" is relatively consistent with ZFC.

In a joint paper with Diego Mejia and Saharon Shelah (still in progress) we construct ultrafilters on N which are, on the one hand, far away from being p-points (there is no Rudin-Keisler quotient which is a p-point), but on the other hand can survive certain forcing iterations adding reals but killing p-points. This shows that non-CH is consistent with small ultrafilter bases AND the nonexistence of p-points. HIFW01 28th August 2015
11:30 to 12:30
L Zdomskyy Delta_1-definability of the non-stationary ideal
The talk will be devoted to the proof of the fact that assuming $V = L$, for every successor cardinal $\kappa$ there exists a GCH and cardinal preserving forcing poset $P \in L$ such that in $L^P$ the ideal of all non-stationary subsets of $\kappa$ is $\Delta_1$-definable over $H(\kappa^+)$. We shall also discuss the situation for limit $\kappa$. HIFW01 28th August 2015
13:30 to 14:00
M Staniszewski On ideal equal convergence
We consider ideal equal convergence of a sequence of functions. This is a generalization of equal convergence introduced by Cs\'{a}sz\'{a}r and Laczkovich. The independent, equivalent definition was introduced by Bukovsk{\'a}. She called it quasi-normal convergence. We study relationships between ideal equal convergence and various kinds of ideal convergences of sequences of real functions.

We prove a characterization showing when the ideal pointwise convergence does not imply the ideal equal (aka quasi-normal) convergence. The characterization is expressed in terms of a cardinal coefficient related to the bounding number. Furthermore we consider ideal version of the bounding number on sets from coideals. HIFW01 28th August 2015
14:00 to 14:30
The size of conjugacy classes of automorphism groups
The automorphism groups of Fraissé limits are usually interesting objects both from group theoretic and set theoretic viewpoint. However, these groups are often non-locally compact, hence there is no natural translation invariant measure on them. Christensen introduced the notion of Haar null sets in non-locally compact Polish groups which is a well-behaved generalisation of the null ideal to such groups. In my talk I will present some new results concerning the size of the conjugacy classes of automorphism groups of Fraisse limits with respect to this notion. HIFW01 28th August 2015
15:00 to 16:00
Y Zhu The higher sharp
We establish the descriptive set theoretic representation of the mouse $M_n^\#$, which is called $0^{(n+1)\#}$. At even levels, $0^{(2n)\#}$ is the higher level analog of Kleene's O; at odd levels, $0^{(2n+1)\#}$ is the unique iterable remarkable level-$(2n+1)$ blueprint. HIFW01 28th August 2015
16:00 to 17:00
Indestructible remarkable cardinals
In 2000, Schindler introduced remarkable cardinals and showed that the existence of a remarkable cardinal is equiconsistent with the assertion that the theory of $L(\mathbb R)$ is absolute for proper forcing. Remarkable cardinals can be thought of either as a miniature version of strong cardinals or as having aspects of generic supercompactness, but they are relatively low in the large cardinal hierarchy. They are downward absolute to $L$ and lie (consistency-wise) between the 1-iterable and 2-iterable cardinals of the $\alpha$-iterable cardinals hierarchy (below Ramsey cardinals). I will discuss the indestructibility properties of remarkable cardinals, which are similar to those of strong cardinals. I will show that a remarkable cardinal $\kappa$ can be made simultaneously indestructible by all $\lt\kappa$-closed $\leq\kappa$-distributive forcing and by all forcing of the form ${\rm Add}(\kappa,\theta)*\mathbb R$, where $\mathbb R$ is forced to be $\lt\kappa$-closed and $\leq\kappa$-distributive. For this argument, I will introduce the notion of a remarkable Laver function and show that every remarkable cardinal has one. Although, the existence of Laver-like functions can be forced for most large cardinals, few, such as strong, supercompact, and extendible cardinals, have them outright. The established indestructibility can be used to show, for instance, that any consistent continuum pattern on the regular cardinals can be realized above a remarkable cardinal and that a remarkable cardinal need not be even weakly compact in ${\rm HOD}$. This is joint work with Yong Cheng. 