Seminars (HIF)

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.

A generalisation of stationarity, associated with stationary reflection, was introduced in [1]. 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 [1] .

If there is time I will also introduce a generalisation of ineffable cardinals and a weak $\diamond$ principal that is associated.

[1] J. Bagaria, M. Magidor, and H. Sakai. Reflection and indescribability in the constructible universe. $\textit{Israel Journal of Mathematics}$, to appear (2012).

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.

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.

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 [1] provide a general framework that unifies several Ramsey type results.

[1] Todorcevic, S. Introduction to Ramsey spaces. Princeton University Press, 2010.

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.

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.

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.

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.

Preamble: In this work we deal with partition relations with infinite exponents under $ZFC$, hence all results are limited to definable functions.

In [78], M. Spector has proven, basically, that $\exists \kappa(o(\kappa) = 1)$ is equiconsistent with $\aleph_1 \rightarrow (\omega)^{\omega}_{\aleph_0}$. In [87], 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 $\alpha$ [87]: $\exists \kappa(o(\kappa) = \alpha)$ 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:

[78] M. Spector: Natural Sentences of Mathematics which are independent of $V = L$, $V = L^\mu$ etc., 1978 (preprint).

[87] Y.M. Kimchi: Dissertation, 1987, The Hebrew University of Jerusalem, Israel

On Mathias Day, the life work of Adrian Mathias in set theory will be surveyed and celebrated, in full range and extent.

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$.

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.

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.

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.

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 $\kappa$.

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.

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.

The modern mathematical story of infinity began in the period 1879-84 with a series of papers by Cantor that defined the fundamental framework of the subject. Within 40 years the key ZFC axioms for Set Theory were in place and the stage was set for the detailed development of transfinite mathematics, or so it seemed. However, in a completely unexpected development, Cohen showed in 1963 that even the most basic problem of Set Theory, that of Cantor's Continuum Hypothesis, was not solvable on the basis of the ZFC axioms.

The 50 years since Cohen's work has seen a vast development of Cohen's method and the realization that the occurrence of unsolvable problems is ubiquitous in Set Theory. This arguably challenges the very conception of Cantor on which Set Theory is based.

Thus a fundamental dilemma has emerged. On the one hand, the discovery, also over the last 50 years, of a rich hierarchy axioms of infinity seems to argue that Cantor's conception is fundamentally sound. But on the other hand, the developments of Cohen's method over this same period seem to strongly suggest there can be no preferred extension of the ZFC axioms to a system of axioms that can escape the ramifications of Cohen's method.

But this dilemma was itself based on a misconception and recent discoveries suggest there is a resolution.

We report on ongoing work with Arno Pauly, showing how concepts from computable analysis can be used to shed light and uniformize certain games for classes of functions which have been studied in descriptive set theory, such as Wadge's game for continuous functions, Duparc's eraser game for Baire class 1 functions, and Semmes' tree game for Borel functions.

As an application, for each finite n we obtain a game characterizing the Baire class n of functions.

In this talk I will survey the general theory of Boolean ultrapowers, starting from the beginnings and including many applications and some possible future developments. Also, the set-theoretic approach to Boolean ultrapowers, due to recent work of Hamkins and Seabold, will be discussed.

First developed by Mansfield as a purely algebraic construction, Boolean ultrapowers are a natural generalization of usual power-set ultrapowers. More specifically, I will focus on how some combinatorial properties of a ultrafilter U are related to the realization of types in the resulting Boolean ultrapower. Many results on $\lambda$-regular and $\lambda$-good ultrafilters, mostly due to Keisler, can be generalized to this context. In particular, I will sketch the construction of a $\lambda$-good ultrafilter on the Levy collapsing algebra $\mathrm{Coll}(w, <\lambda)$. In addition to that, I will describe a possible application to Keisler's order on complete theories.

The theory of Weihrauch degrees is about representing classical theorems of analysis in Baire space and comparing their strength (measured as the Weihrauch degree). In this talk, we are exploring a version of this theory for generalized Baire space. The first step in this generalization is that of finding a generalization of R on which we can prove a version of theorems from classical analysis. The first part of the talk will be devoted to the presentation of the construction of an extension of R on which we can prove a version of the Intermediate Value Theorem. In the second part of the talk we will be focusing on generalizing notions from computable analysis. Finally we will show how this new framework can be used to characterize the strength of the generalized version of the version of the Intermediate Value Theorem we presented in the first half of the talk.

The role of the Axiom of Choice in Mathematics has been studied extensively. Given a theorem of ZFC, one may ask how strong it is compared to the Axiom of Choice. Although a large collection of results has been analysed in this way, there are still simple and elegant theorems that offer resistance. One such result is the Nielsen-Schreier theorem, which states that subgroups of free groups are free.

I will introduce recent results that help to establish the strength of Nielsen-Schreier, focussing on the method of representative functions. Then I discuss potential applications of this technique to other algebraic structures admitting a basis, such as free abelian groups and vector spaces.

We prove that the Semi-Stationary Reflection Principle, together with the negation of the Continuum Hypothesis, implies that $\omega_2$ has the Strong Tree Property. Also, we show that SSR implies the negation of $\Box(\lambda, \omega)$ for all regular cardinals $\lambda\geq\omega_2$. This is a joint work with Liuzhen Wu.

The aim of my talk is to highlight a hitherto unknown computational aspect of Nonstandard Analysis. In particular, we provide an algorithm which takes as input the proof of a mathematical theorem from ‘pure’ Nonstandard Analysis, i.e. formulated solely with the nonstandard definitions (of continuity, integration, dif- ferentiability, convergence, compactness, et cetera), and outputs a proof of the as- sociated effective version of the theorem. Intuitively speaking, the effective version of a mathematical theorem is obtained by replacing all its existential quantifiers by functionals computing (in a specific technical sense) the objects claimed to exist. Our algorithm often produces theorems of Bishop’s Constructive Analysis ([2]). The framework for our algorithm is Nelson’s syntactic approach to Nonstandard Analysis, called internal set theory ([4]), and its fragments based on Goedel’s T as introduced in [1]. Finally, we establish that a theorem of Nonstandard Analysis has the same computational content as its ‘highly constructive’ Herbrandisation. Thus, we establish an ‘algorithmic two-way street’ between so-called hard and soft analysis, i.e. between the worlds of numerical and qualitative results.

 

References: [1] Benno van den Berg, Eyvind Briseid, and Pavol Safarik, A functional interpretation for non- standard arithmetic, Ann. Pure Appl. Logic 163 (2012), no. 12, 1962–1994. [2] Errett Bishop and Douglas S. Bridges, Constructive analysis, Grundlehren der Mathematis- chen Wissenschaften, vol. 279, Springer-Verlag, Berlin, 1985. [3] Fernando Ferreira and Jaime Gaspar, Nonstandardness and the bounded functional interpre- tation, Ann. Pure Appl. Logic 166 (2015), no. 6, 701–712. [4] Edward Nelson, Internal set theory: a new approach to nonstandard analysis, Bull. Amer. Math. Soc. 83 (1977), no. 6, 1165–1198. [5] Stephen G. Simpson, Subsystems of second order arithmetic, 2nd ed., Perspectives in Logic, CUP, 2009.

There exists a fairly tight fit between type theories à la Martin-Löf and constructive set theories such as CZF and its extension, and there are connections to classical Kripke-Platek set theory and extensions thereof, too. The technology for determining the (exact) proof-theoretic strength of such theories was developed in the late 20th century. The situation is rather different when it comes to type theories (with universes) having the impredicative type of propositions Prop from the Calculus of Constructions that features in some powerful proof assistants. Aczel's sets-as-types interpretation into these type theories gives rise to rather unusual set-theoretic axioms: negative power set and negative separation. But it is not known how to determine the consistency strength of intuitionistic set theories with such axioms via familiar classical set theories (though it is not difficult to see that ZFC plus infinitely many inaccessibles provides an upper bound). The first part of the talk will be a survey of known results from this area. The second part will be concerned with the rather special computational and proof-theoretic behavior of such theories.

The aim of this talk is to propose a topic of study that connects infinite matroids with pushdown automata on words indexed by arbitrary linear orders. The motivation for this study is the key open conjecture concerning infinite matroids, the Intersection Conjecture of Nash-Williams, as well as a result from my paper "Infinite Matroidal Version of Hall's Matching Theorem, J. London Math. Soc., (2) 71 (2005), 563–578." The main result of this paper can be described using pushdown automata as follows. Let P=(M,W) be a pair of matroids on the same groundset E. We assign to P a language L_P consisting of transfinite words (indexed by ordinals) on the alphabet A={-1,0,1}. The language L_P is obtained by taking all injective transfinite sequences of the elements of E and translating each such sequence f into a word of L_P. The translation involves replacing an element of f by -1, 0 or 1 depending on whether it is spanned by its predecessors in both, one or none of the matroids M and W.

Theorem There exists a pushdown automaton T on transfinite sequences in the alphabet A such that the language L_T consisting of words accepted by T has the following property: For every pair P of matroids satisfying property (*), the language L_P is a subset of L_T if and only if the pair P has a packing (the ground set E can be partitioned into sets E_M and E_N that are spanning in M and N, respectively).

The property (*) is that M is either finitary or a countable union of finite co-rank matroids and W is finitary.

Co-author: Philipp Schlicht (Universität Bonn)

Transfinite machine models of computation provide an approach to an `effective mathematics of the uncountable'. However, their set-theoretical interest seems to be limited by the fact that even the strongest such model, Koepke's Ordinal Turing Machines with parameters (pOTMs), can only compute constructible sets.

Recognizability is a more liberal notion than computability in that it only requires the machine to be able to identify a certain object when it is given to it as an input, not to produce that object.

By invoking notions from algorithmic randomness and considering recognizability rather than computability, we connect transfinite computability to large cardinals and forcing axioms incompatible with the axiom of constructibility on the one hand and inner models for large cardinals on the other. In particular, under appropriate large cardinal assumptions, a real number is heriditarily recognizable by a pOTM if and only if it is an element of the mouse for one Woodin cardinal. This is joint work with Philipp Schlicht.

We review some recent results on links between (descriptive) set theory and automata theory. In particular, we consider the topological complexity of languages of infinite words accepted by various kinds of automata, the infinite games specified by automata, and independence results in automata theory.

For various questions in Infinite Graph Theory, matroids have turned out to be the right tool to tackle them. This introduction to infinite matroids will be self-contained; in particular I will explain what a matroid is.

Co-authors: Neil Hindman (Howard University), Dona Strauss (University of Leeds)

We show (assuming CH) that there is a finite colouring of the reals such that no infinite set X has X+X (meaning the pairwise sums from X, allowing repetition) monochromatic. And we give positive results for ‘nice’ colourings.

We discuss several results related to the question of when a Borel graph has a Borel matching. Here, the analogue of Hall's matching theorem fails, but there are positive results giving Borel matchings in several contexts if we are willing to discard null or meager sets. We also discuss some applications to geometrical paradoxes. This is joint work with Spencer Unger.