The INFTY Final Conference marks the conclusion of the European Science Foundation Research Networking Programme INFTY – New frontiers of infinity: mathematical, philosophical, and computational prospects. The INFTY network is devoted to the study of a wide range of aspects of Infinity. It principally addresses set theory as the mathematical study of infinity, and applications of set theory within pure mathematics.

The INFTY Final Conference will take place at the Mathematics Center, University of **Bonn**, from Tuesday, March 4 to Friday, March 7, 2014. The scientific program of the conference features invited on set theory and neighbouring areas by internationally leading experts and young researchers as well as contributed talks. A Hausdorff Afternoon will commemorate the publication of Felix Hausdorff’s groundbreaking work Grundzüge der Mengenlehre exactly 100 years ago.

Invited Speakers:

- Arthur Apter (New York)
- Joan Bagaria (Barcelona)
- Dana Bartosova (Sao Paulo)
- Andreas Blass (Ann Arbor)
- Carlos Di Prisco (Carcas)
- Vera Fischer (Vienna)
- Stefan Geschke (Hamburg)
- Michael Hrusak (Morelia)
- Daisuke Ikegami (Torino)
- Juliette Kennedy (Helsinki)
- Peter Koellner (Cambridge USA)
- Miloš Kurilić (Novi Sad)
- Pierre Matet (Caen)
- Luca Motto Ros (Freiburg)
- Walter Purkert (Bonn)
- Assaf Rinot (Ramat-Gan)

Online registration is now possible. The deadline for registration and submission of abstracts is February 17, 2014. The registration fee is 100 EUR for tenured faculty and 50 EUR otherwise.

Please see this site for more information or email inftyfinal@gmail.com.

Some travel and accommodation funding is available for junior participants.

The program committee consists of: Joan Bagaria, ICREA, Barcelona; Mirna Dzamonja, Norwich (Chair); Sy Friedman, U Vienna; Peter Koepke, Bonn; Benedikt Löwe, Amsterdam/Hamburg; Erik Palmgren, Uppsala; Grzegorz Plebanek, Wroclaw; Peter Simon, Prague; Jouko Väänänen, Amsterdam/Helsinki; Andreas Weiermann, Gent.

The organizing committee consists of: Peter Koepke, Philipp Lücke, and Philipp Schlicht.

The Conference is made possible by the generous support of the Hausdorff Center for Mathematics, and by the Infty Network.

## Tuesday, March 4

08:00 – 09:00 |
Registration |

09:00 |
Opening |

09:15 – 10:15 |
Assaf Rinot: Same Graph, Different Universe |

10:15 – 10:45 |
break |

10:45 – 11:30 |
Stefan Geschke: An invitation to infinite matroids |

11:30 – 12:15 |
Luca Motto Ros: Towards a descriptive set theory for computer science |

12:15 – 14:15 |
lunch break |

14:15 – 15:15 |
Arthur Apter: Normal Measures and Strongly Compact Cardinals |

15:15 – 15:45 |
break |

15:45 – 16:15 |
Contributed: Merlin Carl: Infinite Time Algorithmic Randomness (Lipschitzsaal) / Vincenzo Dimonte: I0, Generic Absoluteness and Combinatorics (room 1.008) |

16:15 – 16:45 |
Contributed: Michael Doucha: Scott rank of Polish metric spaces (Lipschitzsaal) / Barnabas Farkas: Almost disjoint refinements (room 1.008) |

16:45 – 17:00 |
break |

17:00 – 17:30 |
Contributed: Yizheng Zhu: Mice from hod mice (Lipschitzsaal) / Yurii Khomskii: Suslin proper forcing and regularity properties (room 1.008) |

## Wednesday, March 5

9:15 – 10:15 |
Andreas Blass: Complete combinatorics, revisited |

10:15 – 10:45 |
break |

10:45 – 11:30 |
Vera Fischer: Template Iterations |

11:30 – 12:15 |
Milos Kurilic: Different Similarities |

12:15 – 14:15 |
lunch break |

14:15 – 14:30 |
Hausdorff afternoon: introduction |

14:30 – 15:30 |
Walter Purkert: Hausdorff’s “Grundzüge der Mengenlehre” – a milestone in the development of modern mathematics |

15:30 – 16:00 |
break |

16:00 – 17:00 |
Juliette Kennedy: On Formalism Freeness |

17:05 – 18:00 |
European Set Theory Society meeting |

## Thursday, March 6

9:15 – 10:15 |
Carlos Di Prisco: Partitions and weak forms of choice |

10:15 – 10:45 |
break |

10:45 – 11:30 |
Dana Bartosova: Lelek fan from a projective Fraïssé limit |

11:30 – 12:30 |
Michael Hrusak: Strong measure zero in separable metric spaces and groups |

12:30 – 18:00 |
Social afternoon |

19:00 |
Conference Dinner (Da Capo restaurant near Beethovenhalle) |

## Friday, March 7

9:15 – 10:00 |
Peter Koellner: An Inconsistency in the Large Cardinal Hierarchy? |

10:15 – 10:45 |
break |

10:45 – 11:30 |
Daisuke Ikegami: MM + + and the generic nice UBH |

11:30 – 12:00 |
Contributed: Borisa Kuzeljevic: On the structure of countable ultrahomogeneous partial orders (Lipschitzsaal) / Sandra Uhlenbrock: Producing models with woodin cardinals from optimal determinacy hypotheses (room 1.008) |

12:00 – 12:30 |
Contributed: Robert Lubarsky: Extensions of supercompactness (Lipschitzsaal) / Edoardo Rivello: Orbit-theoretic revision (room 1.008) |

12:30 – 14:30 |
lunch break |

14:30 – 15:15 |
Pierre Matet: Pκ(λ) versions of results of Shelah on diamond |

15:15 – 15:30 |
break |

15:30 – 16:30 |
Joan Bagaria: Adding a random real number and its effect on Martin’s axiom |

## Abstracts:

#### Arthur Apter: Normal Measures and Strongly Compact Cardinals

I will discuss the question of the possible number of normal measures a non-(κ + 2)-strong strongly compact cardinal κ can carry. This is part of a joint project with James Cummings.

#### Joan Bagaria: Adding a random real number and its effect on Martin’s axiom

In a joint work with Saharon Shelah we show that adding a random real number destroys a large fragment of Martin’s axiom, namely Martin’s axiom for partial orders that have precalibre- ℵ1 , thus answering an old question of J. Roitman. We also answer a question of J. Steprans and S. Watson by showing that, by a forcing that preserves cardinals, one can destroy the precalibre-ℵ1 property of a partial ordering while preserving its ccc-ness.

#### Dana Bartosova: Lelek fan from a projective Fraïssé limit

We show how to construct the Lelek fan as a natural quotient of a projective Fraïssé limit of a certain class of finite trees. We investigate properties of the Lelek fan (such as projective universality and homogeneity) as well as of its automorphism group (among others simplicity and connectedness properties).

#### Andreas Blass: Complete combinatorics, revisited

I plan to first review results, dating back to the 1970’s, concerning complete combinatorics for certain ultrafilters and related partition properties. Then I’ll discuss newer results, about other sorts of ultrafilters, including one sort that recently played a role in the study of the Tukey ordering.

#### Carlos Di Prisco: Partitions and weak forms of choice

We will survey a series of results regarding colorings of sets of natural numbers, often relating partition properties resulting from them with weak forms of the axiom of choice. We will put special interest in weakenings of the Ramsey property, for example considering only colorings of the infinite sets of natural numbers which are invariant under finite changes, or colorings that are not invariant but cyclic. Some of the results that will be presented are part of work done in collaboration with F. Galindo, J. Henle and A. R. D. Mathias.

#### Vera Fischer: Template Iterations

J. Brendle used Shelah’s template iteration techniques to show that the minimal size of a maximal almost disjoint family can be of countable cofinality. Jointly with A. Törnquist we elaborate on the techniques of Shelah and Brendle, and show that the minimal size of a maximal cofinitary group can be of countable cofinality. Our approach is sufficiently general and allows us to establish in addition, that the minimal size of a maximal family of almost disjoint permutations, as well as the minimal size of a maximal eventually different family can also be of countable cofinality.

#### Stefan Geschke: An invitation to infinite matroids

Matroids provide an abstract notion of independence. Finite matroids have been studied for a long time and play a major role in combinatorics and optimization. A satisfactory generalization to the infinite has been obtained only recently. We discuss some set-theoretic issues that arise in the theory of infinite matroids.

#### Michael Hrusak: Strong measure zero in separable metric spaces and groups

We will discuss strong measure zero in the context of Polish groups and separable metric spaces in general.

#### Daisuke Ikegami: MM + + and the generic nice UBH

It is a long-standing open question in inner model theory whether the nice Unique Branch Hypothesis (nice UBH) is provable in ZFC. The generic nice UBH, which states that the nice UBH holds in any set generic extension, has several interesting consequences in inner model theory, generic absoluteness, and forcing axioms. In this talk, assuming the generic nice UBH in V, we find a suitable extension of ZFC + MM + + + large cardinals which is complete to the 1st-order statements in Hω2. This is joint work with Matteo Viale.

#### Juliette Kennedy: On Formalism Freeness

In his 1946 Princeton Bicentennial Lecture Goedel suggested the problem of finding a notion of definability for set theory which is “formalism free” in a sense analogous to the notion of computable function –- a notion which is very robust with respect to its various associated formalisms. One way to interpret this suggestion is to consider standard notions of definability in set theory, which are usually built over first order logic, and change the underlying logic. We show that constructibility is not very sensitive to the underlying logic, and the same goes for hereditary ordinal definability (or HOD). There are also some interesting intermediate models. This is joint work with Menachem Magidor and Jouko Väänänen.

#### Peter Koellner: An Inconsistency in the Large Cardinal Hierarchy?

The hierarchy of large cardinals provides us with a canonical means to climb the hierarchy of consistency strength. There have been many 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 “choiceless” 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.

#### Milos Kurilic: Different Similarities

We consider several “similarity relations” between relational structures. For example, the basic similarity is the equality and the next natural similarity is the isomorphism, which implies one more similarity relation, the equimorphism. Further, we can extend the list of similarities of relational structures comparing the corresponding posets of copies (P(X),⊂) (where P(X) = {f[X]: f ∈ Emb(X)}, for a structure X) which can be equal, isomorphic, forcing equivalent etc. Clearly, all such similarities are equivalence relations on the class of models of a relational language and some results concerning the hierarchy of these equivalences and the corresponding classifications of structures will be presented.

#### Pierre Matet: Pκ(λ) versions of results of Shelah on diamond

We recall two results, one by Gregory and Shelah on diamond star, and the other by Shelah on diamond at a successor cardinal, and discuss generalizations to P_{κ}(λ).

#### Luca Motto Ros: Towards a descriptive set theory for computer science

Wadge-like reductions on quasi-Polish spaces Abstract: In contrast with the theory of terminating computations, the study of infinite computations seems to crucially depend on topological considerations and on the analysis of the complexity of subsets of certain (often nonmetrizable) topological spaces, like continuous domains. Despite the fact that the usual results and methods of descriptive set theory do not readily apply to these spaces because of nonmetrizability, it has recently been shown that a relevant fragment of such theory can be developed in a unified way for the wider collection of the so-called quasi-Polish spaces, a class including both Polish spaces and omega-continuous domains. To illustrate this somewhat surprising phenomenon, we consider the case of the study of Wadge-like reducibilities on arbitrary quasi-Polish spaces.

#### Walter Purkert: Hausdorff’s “Grundzüge der Mengenlehre” – a milestone in the development of modern mathematics

Felix Hausdorff (1868-1942) was not only a versatile mathematician but also a philosopher and a remarkable man of letters. He started his career in applied mathematics with papers on astronomy, optics, probability and insurance mathematics. His philosophical interests led him to begin studying Cantor’s ideas. Set theory soon became his principal field of research. From 1901 to 1909 he achieved fundamental results on ordered sets. In 1912 he started writing his ‘opus magnum’ “Grundzüge der Mengenlehre” which appeared in April 1914, almost right on the day 100 years ago. Set theory at that time included not just the general theory of sets but also point sets as well as the theory of content and measure. Hausdorff’s work was the first textbook that dealt systematically with all aspects of set theory in this comprehensive sense and which provided complete proofs in a masterful form. The talk will give an overview of the book’s content and especially of the significant original contributions by its author including some of the earlier results in the theory of ordered sets. We also deal with the reception of “Grundzüge” and with its influence on the development of modern mathematics. Finally we shortly go into Hausdorff’s tragic fate under the Nazi dictatorship.

#### Assaf Rinot: Same Graph, Different Universe

In a paper from 1998, answering a question of Hajnal, Soukup proved that ZFC+GCH is consistent with the existence of two graphs G,H of size and chromatic number κ = ω_{2}, whose product G*H is countably chromatic. The consistency of the statement for cardinals κ>ω_{2} remained open until recently, where we demonstrated that in Gödel’s constructible universe, this holds simultaneously for every successor cardinal κ. The key idea is the construction of two graphs G and H of size κ , and two <κ-distributive notions of forcing P and Q such that:

(1) in L^{P} , chr(G) = κ , chr(H) = ω

(2) in L^{Q} , chr(G) = ω , chr(H) = κ

Thus, in this talk we shall address the following general question: Given a fixed graph G in a fixed universe V, what are the possible values for chr(G) among all cofinality-preserving forcing extensions of V?

## Contributed Talks

#### Merlin Carl: Infinite Time Algorithmic Randomness

We consider algorithmic randomness for machine models of infinitary computations. We show that a theorem of Sacks, according to which a real x is computable from a randomly chosen real with positive probability iff it is recursive holds for many of these models, but is independent from ZFC for ordinal Turing machines. Furthermore, we define an analogue of ML-randomness for Infinite Time Register Machines and show that some classical results like van Lambalgen’s theorem continue to hold.

#### Vincenzo Dimonte: I0, Generic Absoluteness and Combinatorics

The talk is a recap of recent applications of the Generic Absoluteness Theorem, that extend considerably the known compatibilities of I1 with various combinatorial properties, like the tree property, the failure of SCH or the existence of certain scales.

#### Michael Doucha: Scott rank of Polish metric spaces

We shall present the Scott analysis of Polish metric spaces which has been a subject of study of Fokina, Friedman, Korwien and Nies. We will provide the answer to their question whether the Scott rank of an arbitrary Polish metric space is countable.

#### Barnabas Farkas: Almost disjoint refinements

I am going to prove the following generalization of a result due to J. Brendle: If I is an analytic or coanalytic ideal on the natural numbers and a forcing notion P adds new reals then, in the extension, there is an I-almost disjoint refinement of the family of I-positive sets from the ground model.

#### Yurii Khomskii: Suslin proper forcing and regularity properties

Recently, in joint work with Vera Fischer and Sy Friedman, we obtained a number of new results separating regularity properties on levels of the projective hierarchy above the second level. In this talk I will present some methods that we used to obtain these results, particularly concerning “Suslin and Suslin+ proper forcing”, a strengthening of Shelah’s concept of proper forcing which only works for easily definable forcing posets.

#### Borisa Kuzeljevic: On the structure of countable ultrahomogeneous partial orders

We prove that if P is a countable ultrahomogeneous partial order different from countable antichain, then the order types of maximal chains in the poset of isomorphic suborders of P enlarged with the empty set are characterized as the order types of compact sets of reals with non-isolated minimum. In the case of countable antichain these order types are exactly order types of nowhere dense compact sets of reals with minimum non-isolated. This is joint work with Milos Kurilic.

#### Robert Lubarsky: Extensions of supercompactness

Several large cardinal properties that seem to be slight extensions of supercompactness have been identified in recent years. This talk is about their consistency strength.

#### Edoardo Rivello: Orbit-theoretic revision

Moving from a leitmotiv in Hausdorff’s 1914 *Grundzüge*– the role played by the ordinals in proving mathematical statements – I review Kuratowski’s method of eliminating the ordinals from mathematical reasoning, and extend it to encompass more general situations, known in the literature as “revision constructions”. In particular, I prove two generalisations of Kuratowski’s theorem which apply to the most common revision-theoretic proposals in theories of truth: Herzberger and Belnap revision. These generalisations also allow to re-prove, in a choice-free context, old results until now only proved by appealing to some form of the axiom of choice.

#### Yizheng Zhu: Mice from hod mice

HOD mice have much simpler large cardinals than the consistency-wise counterpart of mice. Thus they are a lot easier to analyze. We describe a construction of mice from HOD mice with equal consistency strength. In particular, every model of AD^{ + }below LST (The largest Suslin cardinal is a member of the Solovay sequence) is a derived model of a premouse. The premouse will be as close to a Woodin limit of Woodins as possible. This method suggests that LST is equiconsistent with a Woodin limit of Woodins.