# Combinatorial set theory at the successor of a singular cardinal: a marriage of a forcing axiom and a reflection principle

Lead Research Organisation:
University of East Anglia

Department Name: Mathematics

### Abstract

The subject of this proposal lies within infinite combinatorics, which means combinatorics of inifinite numbers. Such numbers can be used to model processes that last infinitely many steps and therefore to understand the nature of such processes. The research here is fundamental in nature. Its connection with applications is that infinite processes are expected to have a role in building future generations of artifical intelligence and computing equipment. The computing capabilities known now, including the physical computers that we use in everyday life, are based heavily on finite combinatorics. Infinite cardinal numbers come in two groups, regular and singular. Much more is known about the combonatorics of regular cardinals than the combinatorics of the singular ones. Here we study immediate successors of singular cardinals.The specific subject that the proposal is concerned with is the possibility of having a successor of a singular cardinal on which there is both a forcing axiom and reflection principle. This would stand in sharp contrast with what is possible to achieve at successors of regular cardinals. We address the question if it is possible that the cardinal r0, the first infinite cardinal at which Rado's statement holds, can be the successor of a singular cardinal. The background for this research builds on a central question in set theory, the singular cardinal hypothesis SCH and the related question of understanding the combinatorial nature of successors of singular cardinals. The questions are directly connected to the first problem on the Hilbert's list, now over a century old. The proposal brings an entirely novel technology by which to attack the problem.The research hypothesis and objectives are to use a model of the axiom SSF to get r0 the successor of a singular in a generic extension by Radin forcing. Other outcomes we expect from the project are combinatorial results about sucessors of singulars, and we expect to discover these in the case that our model indeed does give r0 is sucessor of a singular, but also in the case that it does not, because in the latter case we will have to understand why such a combination is not possible.The academic beneficiaries of this project are first of all the set-theoretic community, and then more widely the community of mathematical logicians. As the results become settled we expect applications in fields outside of mathematical logic, such as Banach spaces and measure algebras. The PI has a considerable experience of being able to connect advances in set theory with problems stemming from other areas of mathematics.In the UK, high-end set theory is present at Bristol and in the logic group at UEA. Otherwise, it is unfortunately underrepresented nationwide. It is a very active area in other European countries, including Austria, France, Germany and Poland, and in the United States, Israel, Canada and since recently through a considerable national investement, also Australia. The area has been recognised by the award of a large European grant for research networking INFTY, awarded by the European Science Foundation. We expect the results of this research to be of high interest to a large number of top researchers internationally.

### Planned Impact

The non-academic impact of this proposal and the ways of achieveing it can be described in two distinct ways. These are the long term impact and cultural wealth, and enhancing the creative output. This is a research proposal in a fundamental branch of mathematics, in an area that is part of mathematical foundations. Its results will resolve questions that have been known for decades and only now have a chance to be resolved, thanks to the novel technology proposed here. Fundamental science underpins scientific development and is done with a view of long-term economic and practical impact. Due to the inadequacy of physical computers as we know them to resolve the situations brought up with the economic development, it is to be expected that there will be new generations of physical computers that will use thought processes that are more involved than the ones used now. These thought processes will have to come from fundamental sciences, including foundations of mathematics and specifically infinitary combinatorics. The proposal forms part of this kind of investigation. To ensure the benefit from this impact, we shall make the research available in the widest terms, by publishing in excellent journals, presenting at international conferences and distributing over the web sites and email communications. The research will be clearly marked as having been supported by EPSRC. Another impact the research will have is on enhancing the creative output. The research proposed will have a very strong educational impact on a future generation of mathematicians. This is firstly true from the point of view of the PDRA who will be exposed to research topics that are quite far from his/her thesis work. This is true for any PDRA who can be appointed on the project as the topic is completely novel and could not have been covered in previous research exposure. The PDRA will in addition be exposed to the excellent research atmosphere in the School of Mathematics at UEA, which has one of the strongest mathematical logic groups in the country. On the other hand, the presence of the PDRA at the School will greatly enhance the creative output of the research students at UEA (there are six students in logic at the moment, four of which are working with the PI). To ensure the benefits from the impact, we shall continue our very active programme of logic seminars (meeting weekly for two hours), the PI will collaborate closely with the PDRA and will encourage the research students to have close research contact with the PDRA. The research students already have a programme of junior seminars which the PDRA would be encouraged to join.

### Publications

Benkart G
(2015)

*Memories of Mary Ellen Rudin*in Notices of the American Mathematical Society
Borodulin-Nadzieja P
(2013)

*On the isomorphism problem for measures on Boolean algebras*in Journal of Mathematical Analysis and Applications
Cummings J
(2017)

*A framework for forcing constructions at successors of singular cardinals*in Transactions of the American Mathematical Society
CUMMINGS J
(2016)

*SMALL UNIVERSAL FAMILIES OF GRAPHS ON ? ? + 1*in The Journal of Symbolic Logic
Džamonja M
(2015)

*Some Banach spaces added by a Cohen real*in Topology and its Applications
Džamonja M
(2014)

*Isomorphic Universality and the Number of Pairwise Nonisomorphic Models in the Class of Banach Spaces*in Abstract and Applied AnalysisDescription | We have found out that there was a missing point in the SUSIFA forcing axiom, and we have corrected it. In addition, we have provided new applications and opened up a whole new way of thinking of forcing at the successor of a singular cardinal, in particular in the context of the first possible successor of a singular cardinal. There are several additional published outcomes, where we have succeeded to completely classify the Boolean algebras that support a finitely additive uniformly regular measure in the work with Borodulin-Nadzieja, and where we have obtained negative universality results for isomorphisms of Banach spaces of a given density. The results about the successor of a singular have led to a philosophical investigation of the "axiomatic truth", which is a way to understand to what extent the usual axioms ZFC of set theory really model mathematics. Other outcomes include a philosophical paper on the successor of singular and a survey paper on the same, the first now published and the other in print. The work on the project has also led to a successful grant application for a Leverhulme Research Fellowship on "Nice uncountable constructions". Finally, another successful grant application directly related to the research was written to the Isaac Newton Institute and it relates to their HIF programme, held from August to December 2015 and where the PI is one of te organizers. |

Exploitation Route | The mathematical results are clearly of use in the academic context. They have presented at conferences and papers submitted to the leading journals. The philosophical community have become interested in the foundational issues raised by the research and this has led to a collaboration with the Institute of History and Philosophy of Science at the University of Paris 1 and a publication of several papers in philosophy. A workshop HIG was held at the Isaac Newton Institute and it reached audiences in several fields of mathematics and philosophy. On the impact front, the PI has given large number of public talks, has created an open access video course and has taught it in several universities including in developing countries such as Bosnia and Herzegovina and Seenegal. |

Sectors | Creative Economy,Digital/Communication/Information Technologies (including Software),Education,Culture, Heritage, Museums and Collections,Other |

URL | https://www.logiqueconsult.eu |

Description | The findings have been used in creating educational impact, since I have been giving a large number of public oriented talks on higher infinity, where some of these findings figure prominently. I have also created a module for more mathematically aware audiences that can be accessed on open access from my web site. Finally, I have been using this module to teach in universities around the world and raise awareness of mathematical logic and what it can do for the society. |

First Year Of Impact | 2015 |

Sector | Education,Other |

Impact Types | Cultural |

Description | New course in set theory for Master's students at the Sorbonne Lophisc programme |

Geographic Reach | Asia |

Policy Influence Type | Influenced training of practitioners or researchers |

Impact | Students working in philosophy of mathematics are informed of newest foundational issues regarding the singular cardinals. |

Description | Leverhulme Trust Research Fellowship |

Amount | £50,000 (GBP) |

Organisation | The Leverhulme Trust |

Sector | Charity/Non Profit |

Country | United Kingdom |

Start | 05/2014 |

End | 05/2015 |

Description | London Mathematical Society Computer Science Grants |

Amount | £500 (GBP) |

Organisation | London Mathematical Society |

Sector | Academic/University |

Country | United Kingdom |

Start | 11/2015 |

End | 01/2016 |

Title | Forcing methods at the successor of a singular cardinal |

Description | This is a mathematical method to obtain independence results at the successors of singular cardinals. |

Type Of Material | Improvements to research infrastructure |

Year Produced | 2013 |

Provided To Others? | Yes |

Impact | Other researchers are solving questions in the area using our method. |

Description | Collaboration with ENS Cachan |

Organisation | École Normale Supérieure de Cachan |

Country | France |

Sector | Academic/University |

PI Contribution | An article in preparation with Philippe Schnoebelen and Sylvain Schmitz |

Collaborator Contribution | A collaboration on a journal article |

Impact | A preprint of an article |

Start Year | 2014 |

Description | Collaboration with IHPST |

Organisation | Pantheon-Sorbonne University |

Department | Institute of History and Philosophy of Science and Technology (IHPST) |

Country | France |

Sector | Academic/University |

PI Contribution | A joint paper in preparation with Marco Panza. Became an Associate Partner. |

Collaborator Contribution | A joint paper in preparation. |

Impact | A paper in preparation on asymptotic truth. |

Start Year | 2013 |

Description | Collaboration with Piotr Borodulin-Nadzieja |

Organisation | University of Wroclaw |

Department | Institute of Mathematics |

Country | Poland |

Sector | Academic/University |

PI Contribution | Collaboration with Piotr Borodulon-Nadzieja from the University of Wroclaw, Poland |

Collaborator Contribution | A joint paper. |

Impact | The joint paper listed in the publications. |

Start Year | 2009 |