2016

date	speaker	title
2016-01-07	David Schrittesser (University of Copenhagen, Denmark)	Maximal discrete sets with large continuum
In a previous talk at the KGRC, I showed how to construct definable maximal discrete sets in forcing extensions of $L$, in particular in the Sacks and Miller extension. In particular, the existence of such sets is consistent with $V \neq L$. In this talk I shall show the stronger result that the existence of definable discrete sets is consistent with large continuum. In the process, I show an interesting generalization of Galvin's theorem. In particular, this applies to the example of maximal orthogonal families of measures (mofs). One might hope for a simpler way of constructing a mof in a model with large continuum: to find an indestructible such family in $L$. While such an approach is possible e.g. for maximal cofinitary groups, this is impossible for mofs.
2016-01-14	Leszek Kołodziejczyk (University of Warsaw, Poland)	How unprovable is Rabin's decidability theorem?
Rabin's decidability theorem states the decidability of the monadic second order (MSO) theory of two successors, i.e. of the infinite binary tree with the left- and right-successor relations. The MSO theory of two successors is able to express some nontrivial determinacy principles, and most proofs of Rabin's theorem make use of such principles, so it is reasonable to ask whether the theorem could be unprovable in relatively strong axiomatic theories. I will talk about some joint work with Henryk Michalewski in which we attempt to give a reverse-mathematical characterization of the logical strength needed to prove Rabin's theorem. We show that over $ACA_0$, the complementation theorem for nondeterministic tree automata, which is a crucial ingredient of typical proofs of Rabin's theorem, implies a sentence expressing the determinacy of all $Bool({\bf \Sigma^0_2})$ games. Moreover, using results due to MedSalem-Tanaka, Möllerfeld and Heinatsch-Möllerfeld, we show that over $\Pi^1_2$-$CA_0$, this sentence is actually equivalent to Rabin's theorem restricted to the $\Pi^1_3$ fragment of MSO. It follows from our work and from known results on determinacy principles that even a restricted version of Rabin's theorem is unprovable in $\Delta^1_3$-$CA_0$. On the other hand, any version that can be stated in second-order arithmetic is provable in $\Pi^1_3$-$CA_0$ (and, in fact, in $\Pi^1_2$-$CA_0$ plus $\Pi^1_3$ induction).
2016-01-21	Ari Brodsky (Bar-Ilan University, Tel Aviv, Israel)	Custom-made Souslin trees
We propose a parameterized proxy principle from which $\kappa$-Souslin trees with various additional features can be constructed, regardless of the identity of $\kappa$. We then introduce the microscopic approach, which is a simple method for deriving trees from instances of the proxy principle. As a demonstration, we give a construction of a coherent $\kappa$-Souslin tree that applies also for $\kappa$ inaccessible. Here are the [/KGRC-Brodsky-corrected.pdf slides] for this talk (version of 2016-01-27 including minor corrections).
2016-02-11	Keita Yokoyama (Japan Advanced Institute of Science and Technology, Nomi, Ishikawa, Japan and University of Berkeley, California, USA)	The proof-theoretic strength of Ramsey's theorem for pairs and two colors
In the study of reverse mathematics, determining the first-order strength of Ramsey's theorem for pairs and two colors ($RT^2_2$) is a long-term open problem. Hirst showed that $RT^2_2$ implies $\Sigma^0_2$-bounding and Cholak/Jockusch/Slaman showed that $RT^2_2$ is $\Pi^1_1$-conservative over $\Sigma^0_2$-indction. Note that the proof-theoretic strength of $\Sigma^0_2$-bounding is the same as that of $\Sigma^0_1$-induction, so the proof-theoretic strength (or consistency strength) of $RT^2_2$ is in between $\Sigma^0_1$-induction and $\Sigma^0_2$-indction. Recently, the project of deciding the first-order strength of $RT^2_2$ has been strongly carried out using forcing constructions or priority arguments on nonstandard models of $\Sigma^0_2$-bounding mainly by Chong, Slaman and Yang, and they proved in particular that $RT^2_2$ does not imply $\Sigma^0_2$-indction. In this talk, we use a hybrid of forcing construction, indicator arguments, and proof-theoretic technique to show that the $\Pi^0_3$-part of $RT^2_2$ is exactly the same as $\Sigma^0_1$-induction, thus, the proof-theoretic strength of $RT^2_2$ is exactly the same as $\Sigma^0_1$-induction. This is a joint work with Ludovic Patey.
2016-03-03	Yann Pequignot (KGRC)	From well to better, the space of ideals
A well quasi-order (wqo) is a well-founded quasi-order which contains no infinite antichain. The theory of wqos has applications in many contexts and consists essentially of developing tools in order to show that a certain quasi-order suspected to be wqo is indeed so. This theory exhibits a curious and interesting phenomenon: to prove that a certain quasi-order is wqo, it may very well be easier to show that it enjoys a much stronger property. This observation may be seen as a motivation for considering the complicated but ingenious concept of better-quasi-order (bqo) invented by Nash-Williams in 1965. After a motivated introduction to the concept of bqo, I will sketch the proof of a conjecture made by Pouzet in 1978 which states that any wqo whose ideal completion remainder is bqo is actually bqo. The proof relies on a result with both a combinatorial and a topological flavour concerning maps from a front into a compact metric space. This is joint work with Raphaël Carroy.
2016-03-10	Sy-David Friedman (KGRC)	Descriptive Set Theory and Absoluteness
One of the initial motivations for the development of descriptive set theory (Borel-Baire-Lebesgue in Paris, Lusin-Egorov in Moscow) was to avoid the difficuilties of abstract set theory by focusing on sets of reals which have definitions of low complexity. In this talk I'll take a look at the extent to which this idea succeeds in the study of definable equivalence relations. An analytic equivalence relation can have countably-many (small), uncountably-many but not perfectly-many (medium), or perfectly-many classes (large); in the last case it can be either Borel or non-Borel. The classes of an analytic equivalence relation can be countable (small) or contain a perfect set (large). For co-analytic equivalence relations they can also be uncountable with no perfect subset (medium). In either case a large class can either be Borel or non-Borel. I'll discuss the absoluteness/non-absoluteness of these notions as well as some related questions which connect to issues in the theory of class forcing.
2016-03-17	Ben Miller (KGRC)	A basis theorem for the complement of the first Baire class
I will sketch a proof of the fact that there is a six-element basis, under topological embeddability, for the family of Borel functions which are not in the first Baire class. This is joint work with Raphaël Carroy.
2016-04-07	Radek Honzík (Charles University in Prague, Czech Republic)	The tree property and the continuum function below $\aleph_\omega$
We say that a regular cardinal $\kappa \ge \omega$ has the tree property if there are no $\kappa$‑Aronszajn trees. It is known that if $2^\omega = \omega_1$, there are $\omega_2$‑Aronszajn trees; thus the tree property at $\omega_2$ implies the negation of CH (and analogously for larger cardinals). All the usual forcings for the tree property at $\omega_2$, such as the Mitchell forcing or the Sacks forcing, give $2^\omega = \omega_2$. We show that the “gap two” is no consequence of the tree property: indeed, we show that – starting with infinitely many weakly compact cardinals – the tree property can hold at every even cardinal below $\aleph_\omega$ and the continuum function below $\aleph_\omega$ can be arbitrary (such that $2^{\omega_2n} \ge \omega_{2n+2}, n \lt \omega$). We prove a similar result for the weak tree property as well ($\kappa$ has the weak tree property if there are no special $\kappa$‑Aronszajn trees). This work is joint with S. Stejskalova.
2016-04-14	Yue Yang (National University of Singapore, Singapore)	A Normal Form Theorem of Computation on Real Numbers
We define a class of computable functions over real numbers using functional schemes similar to the class of primitive and partial recursive functions defined by Gödel and Kleene. We show that this class of functions can also be characterized by master-slave machines. The proof of the characterization gives a normal form theorem in the style of Kleene; and the equivalence itself illustrates that this characterization is a natural combination of two most influential theories of computation over real numbers, namely, the type-two theory of effectivity (TTE) (see, for example, Weihrauch's book) and the Blum-Shub-Smale model of computation (BSS). This is a joint work with Keng Meng Ng from Nanyang Technological University, Singapore and Nazanin Tavana from Amirkabir University of Technology, Iran.
2016-04-21	Jerzy Kąkol (Adam Mickiewicz University Poznań, Poland)	Selected topics for the weak topology of Banach spaces
Corson (1961) started a systematic study of certain topological properties of the weak topology $w$ of Banach spaces $E$. This line of research provided more general classes such as reflexive Banach spaces, Weakly Compactly Generated Banach spaces and the class of weakly $K$-analytic and weakly $K$-countably determined Banach spaces. On the other hand, various topological properties generalizing metrizability have been studied intensively by topologists and analysts. Let us mention, for example, the first countability, Frechet-Urysohn property, sequentiality, $k$-space property, and countable tightness. Each property (apart the countable tightness) forces a Banach space $E$ to be finite-dimensional, whenever $E$ with the weak topology $w$ is assumed to be a space of the above type. This is a simple consequence of a theorem of Schluchtermann and Wheeler that an infinite-dimensional Banach space is never a $k$-space in the weak topology. These results show also that the question when a Banach space endowed with the weak topology is homeomorphic to a certain fixed model space from the infinite-dimensional topology is very restrictive and motivated specialists to detect the above properties only for some natural classes of subsets of $E$, e.g., balls or bounded subsets of $E$. We collect some classical and recent results of this type, and characterize those Banach spaces $E$ whose unit ball $B_w$ is $k_\mathbb{R}$-space or even has the Ascoli property. Some basic concepts from probability theory and measure theoretic properties of the space $\ell_1$ will be used.
2016-04-28	Jan van Mill (University of Amsterdam, The Netherlands)	The existence of a connected meager in itself CDH space is independent of ZFC
We show that the existence of a countable dense homogeneous metric space which is connected and meager in itself is independent of ZFC. Joint work with Michael Hrusak.
2016-05-12	Anush Tserunyan (University of Illinois at Urbana-Champaign, USA)	Differentiation of subsets of semigroups, a Ramsey theorem, and a van der Corput lemma
A major theme in ergodic Ramsey theory is proving multiple recurrence results for measure-preserving actions of semigroups. What often lies at the heart of these results is that mixing ($\approx$ "chaotic") along a suitable filter on the semigroup amplifies itself to multiple mixing ($\approx$ even more "chaotic") along the same filter. This amplification is usually proved using a so-called van der Corput difference lemma. Instances of this lemma for specific filters have been proven before by Furstenberg, Bergelson–McCutcheon, and others, with a somewhat different proof in each case. We define a notion of differentiation for subsets of semigroups and isolate a class of filters that respect this notion. The filters in this class (call them $\partial$-filters) include all those, for which the van der Corput lemma was known, and our main result is a van der Corput lemma for $\partial$-filters, which thus generalizes its previous instances. This is done via proving a Ramsey theorem for graphs on the semigroup.
2016-05-13	Paul Gartside (University of Pittsburgh, Pennsylvania, USA)	Neighborhoods, Compacta and the Tukey Order
The neighborhoods of a point or set in a topological space form a filter. The compact subsets of a space form a (pre-)filter. Both are directed sets. In this talk we investigate the properties of these directed sets, and the connections between them. The main tool is the Tukey order. We will focus on separable metrizable spaces, and topological objects derived from them, such as function spaces and free topological groups. An important role will be played by chain conditions, in particular calibre $(\omega_1,\omega)$. We will then turn this around and use topological techniques to construct interesting examples of directed sets with and without certain chain conditions.
2016-05-19	Otmar Spinas (Christian-Albrechts-Universität zu Kiel, Schleswig-Holstein, Germany)	Tukey relations between the Mycielski and the meager and null ideals
I have the two results that the meager ideal is Tukey reducible to the Mycielski ideal and that the Lebesgue null ideal consistently is not. In my talk at the KGRC I will explain the main ideas of the proof of the first one, in my Friday [http://dmg.tuwien.ac.at/fg1/SETSeminar/Abstracts/20160520.html talk at the TU] I will give some background of these results and some basic ideas of the proof of the second one, especially how to construct a Silver amoeba that does not add random reals. ''Note: The talk at the [http://dmg.tuwien.ac.at/fg1/Contact.html Set Theory Group] at TU Wien will take place at [https://www.google.at/maps/place/Wiedner+Hauptstra%C3%9Fe+8-10,+1040+Wien/@48.1987104,16.3675761,17z/data=!4m2!3m1!1s0x476d07830d15fc5d:0x84d23ef9c23ca958 Wiedner Hauptstrasse 8-10], green area, 8th floor on 2016-05-20 at 1:00pm.''
2016-06-02	David Chodounský (Academy of Sciences of the Czech Republic, Prague)	Forcing variations of the Martin's Axiom
The classical technique of iterated forcing due the Solovay and Tennenbaum provides a constriction of a "$<c$ universal" c.c.c. forcing, i.e. a forcing such that in the generic extension the Martin's Axiom holds; on c.c.c. posets $<c$ generic filters exist. This motivates the following question. Given a reasonable class of (c.c.c.) forcing notions, does there exist a "$<c$ universal" forcing within this class? I will show the answer to this question is YES, the finite support iteration approach still works, but the reasoning is somewhat more involved than in the classical case. In particular, I will prove that assuming a diamond principle on $\kappa$, given a class of c.c.c. forcings closed on finite support iterations and regular subforcings there is a forcing within this class which forces the Martin Axiom for this class together with the continuum equal to $\kappa$. The talk should be quite basic, I will review the classical method of forcing MA and I will point out the extra challenges in the general setup.
2016-06-09	Sandra Uhlenbrock (Westfälische Wilhelms-Universität Münster, Germany)	Hybrid Mice and Determinacy in the $L(\mathbb{R})$-hierarchy
This talk will be an introduction to inner model theory the at the level of the $L(\mathbb{R})$-hierarchy. It will focus on results connecting inner model theory to the determinacy of certain games. Mice are sufficiently iterable models of set theory. Martin and Steel showed in 1989 that the existence of finitely many Woodin cardinals with a measurable cardinal above them implies that projective determinacy holds. Neeman and Woodin proved a level-by-level connection between mice and projective determinacy. They showed that boldface $\boldsymbol\Pi^1_{n+1}$ determinacy is equivalent to the fact that the mouse $M_n^\#(x)$ exists and is $\omega_1$-iterable for all reals $x$. Following this, we will consider pointclasses in the $L(\mathbb{R})$-hierarchy and show that determinacy for them implies the existence and $\omega_1$-iterability of certain hybrid mice with finitely many Woodin cardinals, which we call $M_k^{\Sigma, \#}$. These hybrid mice are like ordinary mice, but equipped with an iteration strategy for a mouse they are containing, which enables them to capture certain sets of reals. We will discuss what it means for a mouse to capture a set of reals and outline why hybrid mice fulfill this task. If time allows we will sketch a proof that determinacy for sets of reals in the $L(\mathbb{R})$-hierarchy implies the existence of hybrid mice.
2016-06-16	Neil Barton (KGRC)	(Sub)systems of second-order set theory
Much of set-theoretic practice concerns questions that are, at first blush, second-order in content. The study of the construction of inner models (such as Woodin's Ultimate-$L$ conjecture and the construction of the Steel Core Model), the investigation of embedding principles (for example large cardinals and Kunen's Theorem that there is no $j: V \longrightarrow V$), and examination of certain kinds of maximality principle (such as Friedman's Inner Model Hypothesis), are all naturally understood as concerning second-order classes rather than sets. Understandably, given the pleasant metalogical properties of first-order ZFC, many set theorists work hard to render their second-order interests in first-order terms. However, increasingly set theorists have become engaged in questions that are greater than first-order (good examples being the results concerning embeddings from inner models to the universe, the study of open determinacy for class games, and the consistency of the IMH). In the philosophical literature, there is a debate concerning how to characterise proper classes within the framework of there being a unique, maximal proper class model of set theory. Traditionally, talk of proper classes in set theory was understood as shorthand for statements definable in terms of first-order formulae with parameters. However, in the last forty years, philosophical conceptions of proper classes have been proposed which aim to capture this essentially second-order character of set-theoretic practice. In particular, Boolos and Uzquiano develop a paraphrase in terms of plural quantification, where Horsten and Welch provide a mereological conception of proper classes. In this paper, we examine what can be extracted from particular philosophical conceptions of classes, focussing on the plural conception. First ($\S1$), we provide some motivating considerations for the choice of the plural paraphrase. In particular, we argue that the plural paraphrase meshes better with the foundational role many have seen for set theory. Next ($\S2$), we note that this conception of classes has been viewed to motivate one of two class theories, either (1.) MK or (2.) NBG. We argue that this is a false dichotomy; just as in the case of subsystems of second-order arithmetic, we should expect there to be various philosophical motivations for different strengths of class theories both intermediate between NBG and MK, and above MK. Finally ($\S3$), we examine some of the relevant technical literature, and draw some philosophical conclusions. We argue that naturalistic considerations motivate the use of some non-definable class talk. In particular, we argue for two conclusions (1.) $\Pi^1_1$-comprehension for classes is motivated by its having many independently justified consequences made clear in the work of Gitman and Hamkins, and (2.) given a stronger naturalism we can justify the use of strong choice principles for classes extending MK on the basis of work of Gitman. We conclude that a detailed philosophical and mathematical study of (sub)systems of second-order set theory is in order, including some which extend MK.
2016-06-28	Peter Nyikos (University of South Carolina, Columbia, USA)	A forcing built around a coherent Souslin tree and its uses for normal, locally compact spaces
A form of forcing involving ground models with coherent Souslin trees was invented by Paul Larsen and Stevo Todorcevic in order to lay to rest a 1948 problem of Katetov, who had shown that a compact space $X$ is metrizable iff either $X^2$ is perfectly normal or $X^3$ is hereditarily normal (abbreviated $T_5$: this means every subspace is normal). In 1977 I found a nice example if there is a Q-set, of a space $X$ where $X^2$ is $T_5$ but $X$ is not metrizable, and later Gary Gruenhage found a completely different example under CH. Larson and Todorcevic found a model in 2002 where there are no such examples. Their technique consisted of forcing from a ground model with a coherent Souslin tree $S$ to get all ccc posets $P$ that presere $S$ to have filters meeting any collection of $< \mathfrak c$ dense subsets of $P$ [Such models are referred to by the shorthand MA(S).] and then forcing with $S$ itself, resulting in MA(S)[S] models. These models have "paradoxical" properties, satisfying some consequences of V=L such as "every first countable normal space is collectionwise Hausdorff" and some consequences of MA($\omega_1)$ such as "every separable locally compact normal space is hereditarily separable and hereditarily Lindelöf." Since then, the technique has been expanded to replace "ccc" with "proper" to give PFA(S)[S] models and very recently to replace it with "semi-proper" to give MM(S)[S] models. Locally compact spaces of various sorts have been shown to have a host of simplifying properties in these models. One striking recent example: Theorem. In MM(S)[S] models, every locally compact, $T_5$ space is either hereditarily paracompact or contains a copy of the ordinal space $\omega_1$. Many other examples will be surveyed and shown not to follow just from ZFC. For instance, a Souslin tree with the order topology is a (consistent!) counterexample to the topological statement in the preceding theorem.
2016-06-30	Rachid Atmai (KGRC)	Descriptive inner model theory and consistency results in ZFC from determinacy
This talk is about a program that seeks to establish the consistency of combinatorial statements in ZFC from determinacy axioms and which studies the effects of the descriptive set theory in a determinacy model to a universe with choice.
2016-07-14	Arnold W. Miller (University of Wisconsin - Madison, USA)	On the length of Borel hierarchies
I will be giving a survey talk on everything I know about the length of Borel hierarchies.
2016-10-03	Ján Pich (KGRC)	Complexity theory in feasible mathematics
We investigate the provability of conjectures from complexity theory in theories of bounded arithmetic. As we will see, this is closely related to the central problems in proof complexity like lower bounds on the lengths of proofs in Frege systems and, in particular, to Razborov's conjecture about Nisan-Wigderson generators which gives us candidate hard tautologies for strong proof systems. Additionally, we will show that quasi-polynomial circuit lower bounds for SAT are unprovable in the theory $V^0$.
2016-10-13	Daniel Soukup (KGRC)	Some graph theory from the Rockies
The goal of this talk is to describe two projects I have been involved with at the University of Calgary in the last 8 months. First, we will look at the problem of finding independent partial transversals in sparse infinite graphs: we show that if $G$ is an infinite $K_n$-free graph and $m$ is finite then there is a finite $r=r(G,m)$ so that whenever the vertices of $G$ are partitioned into $r$ sets of equal size then there is an independent set $A$ which meets at least $m$ classes in a set of size of $G$. We determine the exact value of $r(G,m)$ for a few examples, in particular for Henson's homogeneous, universal $K_n$-free graphs. Joint work with C. Laflamme, A. A. Lopez, and R. Woodrow. Second, we look at the following conjecture of S. Thomasse: for any digraph $D$, there is an appropriate set $C$ of cycles so that after reversing the orientation along members of $C$ the resulting digraph can be covered by two acyclic vertex sets. We will summarize the current knowledge on this question in the finite and infinite case. Joint work with C. Laflamme and A. A. Lopez.
2016-10-20	Piotr Szewczak (Cardinal Stefan Wyszynski University in Warsaw, Poland)	Products of Menger spaces
A topological space $X$ is Menger if for every sequence of open covers $O_1, O_2,$ $\dots$ there are finite subfamilies $F_1$ of $O_1$, $F_2$ of $O_2,$ $\dots$ such that their union is a cover of $X$. The above property generalizes $\sigma$-compactness. We provide examples of Menger subsets of the real line whose product is not Menger under various set theoretic hypotheses, some being weak portions of the Continuum Hypothesis, and some violating it. The proof method is new.
2016-10-27	Boaz Tsaban (Bar-Ilan University, Tel Aviv, Israel)	On the existence of real Fréchet-Urysohn function spaces
A topological space is Fréchet-Urysohn if every point in the closure of a set is in fact a limit of a sequence in that set. For a real set $X$, let $C(X)$ be the space of continuous real functions, with the topology of pointwise convergence. Gerlits and Nagy proved that the space $C(X)$ is Fréchet-Urysohn if and only if the real set $X$ has a certain elegant covering property. Sets $X$ with this property are called $\gamma$-sets. In 2011, I concluded a line of research with the ultimate possible construction of a concentrated real $\gamma$-set. This resolved a good number of problems in real set theory. Since then, the theorem was verified and applied several times, and refined results were obtained, but no one, including me, really understood the proof. I will present the main ingredients of a more intuitive proof, that grows naturally from the study of related covering properties (known as selection principles). This proof was presented successfully, with full details, in a course for graduate and advanced undergraduate students. ==== Reference ==== T. Orenshtein, B. Tsaban, Linear $\sigma$-additivity and some applications, Transactions of the American Mathematical Society 363 (2011), 3621-3637.
2016-11-03	Raphaël Carroy (KGRC)	Linear orders: when embedding and epimorphism coincide
When a linear order has an order preserving surjection onto each of its suborders, we say that it is strongly surjective. I will mainly talk about countable strongly surjective linear orders, outlining the proof that they form a complete set for the class of unions of an analytic and a coanalytic set. I will then discuss the case of uncountable strongly surjective orders. This is a joint work with Riccardo Camerlo and Alberto Marcone.
2016-11-10	Damian Sobota (KGRC)	On the choice in Rosenthal's lemma
Rosenthal's lemma in its most basic form states that given an infinite matrix $(m_n^k)_{n,k\in\omega}$ of non-negative reals such that $\sum_{n\in\omega}m_n^k\le 1$ for every $k\in\omega$, and $\varepsilon>0$, there exists an infinite set $A\subset\omega$ such that $\sum_{n\in A,n\neq k}m_n^k\le\varepsilon$ for every $k\in A$. The lemma has numerous important applications in Banach space theory and vector measure theory — I will mention some of them during the talk (on the fly explaining and exemplifying all notions and terms). A natural question arises — can the choice of a set $A$ in Rosenthal's lemma be somehow controlled, i.e. can $A$ be chosen from some fixed family $\mathcal{F}\subset[\omega]^\omega$? I will show that it is not possible if $\mathcal{F}$ has cardinality strictly less than $\text{cov}(\mathcal{M})$ (the covering of category). On the other hand, if $\mathcal{F}$ is a basis of a selective ultrafilter (assuming one exists), then $A$ can be chosen from $\mathcal{F}$.
2016-11-17	Andrea Medini (KGRC)	The topology of filters
Any collection $X$ consisting of subsets of $\omega$ can be viewed as a subspace of $2^\omega$ by identifying $X$ with the set of characteristic functions of its elements. This way, it is possible to study the interaction between the set-theoretic and the topological properties of $X$. I will give a survey of results and open questions on this topic, in the case when $X=F$ is a filter on $\omega$.
2016-11-24	Tin Lok Wong (KGRC)	ACT forcing
ACT stands for "Arithmetized Completeness Theorem". The usual proof of Gödel's Completeness Theorem for first-order logic is evidently a forcing-style construction. In many applications, such a construction can easily be transformed into a (complicated perhaps, but natural) recursive construction. I will talk about one example for which this is not the case in the model theory of arithmetic.
2016-12-01	Sandra Uhlenbrock (KGRC)	$\operatorname{HOD}^{M_n(x,g)}$ is a core model
Let $x$ be a real of sufficiently high Turing degree, let $\kappa_x$ be the least inaccessible cardinal in $L[x]$ and let $G$ be $Col(\omega, {<}\kappa_x)$-generic over $L[x]$. Then Woodin has shown that $\operatorname{HOD}^{L[x,G]}$ is a core model, together with a fragment of its own iteration strategy. Our plan is to extend this result to mice which have finitely many Woodin cardinals. We will introduce a direct limit system of mice due to Grigor Sargsyan and sketch a scenario to show the following result. Let $n \geq 1$ and let $x$ again be a real of sufficiently high Turing degree. Let $\kappa_x$ be the least inaccessible strong cutpoint cardinal of $M_n(x)$ such that $\kappa_x$ is a limit of strong cutpoint cardinals in $M_n(x)$ and let $g$ be $Col(\omega, {<}\kappa_x)$-generic over $M_n(x)$. Then $\operatorname{HOD}^{M_n(x,g)}$ is again a core model, together with a fragment of its own iteration strategy. This is joint work with Grigor Sargsyan.