Periodic Reporting for period 1 - IMIC (Inner models and infinite computations)
Reporting period: 2018-07-01 to 2020-06-30
Descriptive set theory is a classical topic that dates back to the 19th century. It studies simply definable sets and functions on complete metric spaces, for instance Borel, analytic and projective sets. A typical question is whether sets of a certain complexity are Lebesgue measurable. There are subtle logical issues here, for instance, strong axioms of infinity beyond the axioms of set theory are necessary to show that all projective sets are Lebesgue measurable. In the context of descriptive set theory, it is useful to understand reals, or elements of some other complete metric space, as infinite words (of length N) whose entries are natural numbers. Measurability, and other similar properties, can then be formulated via infinite games, where each move is a natural number. This approach was used extensively by Kechris and others.
The objective of this project is to study the descriptive set theory of uncountable words from the viewpoints of set theory and higher computability. Spaces of such words are called generalised Baire spaces. The study of these spaces was initiated by Väänänen and has become a major research theme only in the last decade. Major challenges appear because of fundamental differences to the countable setting, caused by combinatorics of uncountable cardinals, for example the existence of Kurepa trees.
One approach links set theory with higher computability. We study uncountable words that can be detected by an algorithm. An analysis of countable recognisable words was carried out in work of Carl, Schlicht and Welch. Hamkins, Leahy and Groszek proved results on the related notion of implicitly definable sets. We aim to understand the precise nature of uncountable recognisable sets, using tools in inner model theory and large cardinals.
Another approach lies in applications of games of uncountable length to definable subsets of generalised Baire spaces. Previous results show that it is consistent that some natural games of uncountable length are determined, for instance the perfect set game and the Banach-Mazur game. We aim to study games for analogues to classical dichotomies and other important games such as Lipschitz games in the uncountable setting.
The objective of this project is to study the descriptive set theory of uncountable words from the viewpoints of set theory and higher computability. Spaces of such words are called generalised Baire spaces. The study of these spaces was initiated by Väänänen and has become a major research theme only in the last decade. Major challenges appear because of fundamental differences to the countable setting, caused by combinatorics of uncountable cardinals, for example the existence of Kurepa trees.
One approach links set theory with higher computability. We study uncountable words that can be detected by an algorithm. An analysis of countable recognisable words was carried out in work of Carl, Schlicht and Welch. Hamkins, Leahy and Groszek proved results on the related notion of implicitly definable sets. We aim to understand the precise nature of uncountable recognisable sets, using tools in inner model theory and large cardinals.
Another approach lies in applications of games of uncountable length to definable subsets of generalised Baire spaces. Previous results show that it is consistent that some natural games of uncountable length are determined, for instance the perfect set game and the Banach-Mazur game. We aim to study games for analogues to classical dichotomies and other important games such as Lipschitz games in the uncountable setting.
A main goal was to determine whether the recognisable sets of ordinals are computable from a canonical inner model. We obtained a positive answer for subsets of the first uncountable cardinal. Strong axiomatic hypotheses on the existence of large cardinals were used. For the second uncountable cardinal, we obtained a negative answer via a different approach using forcing axioms. The difference between the cases was unexpected and opens up new research problems.
An analysis of recognisable sets was also carried out in the first critical case, the least inner model with infinitely many measurable cardinals. This work built on work of Steel and Welch combined with results of Dehornoy on iterated ultrapowers and forcing.
In generalised descriptive set theory, we focused on applications of games of uncountable length. A main achievement is the consistency of an analogue to the infinite dimensional open graph dichotomy of Carroy, Miller and Soukup, and the determinacy of the associated game, in joint work with Sziraki.
A number of closely related problems on the interface of descriptive set theory, forcing, computability and theoretical computer science were studied within the project. The results led to several publications, preprints and talks in international conferences and seminars.
An analysis of recognisable sets was also carried out in the first critical case, the least inner model with infinitely many measurable cardinals. This work built on work of Steel and Welch combined with results of Dehornoy on iterated ultrapowers and forcing.
In generalised descriptive set theory, we focused on applications of games of uncountable length. A main achievement is the consistency of an analogue to the infinite dimensional open graph dichotomy of Carroy, Miller and Soukup, and the determinacy of the associated game, in joint work with Sziraki.
A number of closely related problems on the interface of descriptive set theory, forcing, computability and theoretical computer science were studied within the project. The results led to several publications, preprints and talks in international conferences and seminars.
Although this project is in set theory, it has connections with higher computability and theoretical computer science. This has an impact by fostering dialogue between these fields. Connections between set theory and computability have also recently gained momentum in other work, for instance of Greenberg and Marks. Within set theory, the project combined combinatorics, forcing and large cardinals in novel ways and led to new collaborations.
The results have already led to applications in generalised descriptive set theory, opened up new avenues of research and some ideas have been followed up by other researchers. Our work on recognisable sets is closely linked to very recent work of Koellner and Woodin on categorical theories for transitive models of countable height, and we expect that our results will be useful to study models of uncountable height. In the interface of descriptive set theory and higher computability, our work led to a conjecture that links the theorem of Mansfield and Solovay with a celebrated result of Louveau that has also come up in recent work of Kihara.
I would finally like to mention that this project was crucial for my research output over the last two years, and I am thankful for this unique opportunity.
The results have already led to applications in generalised descriptive set theory, opened up new avenues of research and some ideas have been followed up by other researchers. Our work on recognisable sets is closely linked to very recent work of Koellner and Woodin on categorical theories for transitive models of countable height, and we expect that our results will be useful to study models of uncountable height. In the interface of descriptive set theory and higher computability, our work led to a conjecture that links the theorem of Mansfield and Solovay with a celebrated result of Louveau that has also come up in recent work of Kihara.
I would finally like to mention that this project was crucial for my research output over the last two years, and I am thankful for this unique opportunity.