Topics in Descriptive Set Theory
Lead Research Organisation:
University of Bristol
Department Name: Mathematics
Abstract
We analyse higher types of recursion, non-monotonic definitions and inner model theory to build a new approach to low level determinacy.
People |
ORCID iD |
| Philip Welch (Primary Supervisor) | |
| Christopher Turner (Student) |
| Description | Using the grant (and a short extension to cover some Covid-related delays), I discovered enough new mathematics to receive my PhD. Specifically, I made significant discoveries in 3 areas of set theory over the course of the grant. 1) I discovered a new, more general, formulation of classical forcing axioms, known as name principles. Forcing axioms are a very common tool for set theorists, especially in recent years, but they are arcane to work with. Name principles simplify things a great deal. They allow many proofs to be simplified or improved, as well as opening a new subfield investigating Name Principles themselves. This work was collaborative with Philipp Schlicht, and resulted in a 50 page paper which has been accepted by the Annals of Pure and Applied Logic. 2) Working with Philip Welch, I discovered new results in a field of set theory known as "inner model theory". I discovered a process whereby we can take a very small object (known, for various reasons, as a "machete mouse") and use it to build certain very large structures which are of mathematical interest. I also showed that all of these "machete mice" will be found within another, related, interesting structure. This chapter of my thesis has been turned into a paper, which I am currently editing to incorporate some improvements that have since noticed. 3) I discovered new results about Lowenheim-Skolem-Tarski numbers. These sit on the border between set theory and the (related) field of model theory. An LST number numerically quantifies how complex a mathematical concept is. I found lower bounds for the LST numbers of several interesting concepts, and also showed that - assuming a certain well-known hypothesis is true - this lower bound cannot be improved. Again, this work turned into a paper, which is currently in preprint. In addition to this, I also collaborated on a project investigating infinite games, which resulted in another paper which was accepted by the Bulletin of Symbolic Logic. |
| Exploitation Route | Being a project in pure mathematics, the main users of my outcomes will be others in academia. In the past, set theorists often proved simple name principles ad-hoc, but did not realise how far this approach could go; my results allow many known results to be improved, and to have their proofs greatly simplified. It is also likely there are many brand new results that can now be discovered, which were inaccessible before. My LST number work, meanwhile, is already being put to use by at least one PhD student in Hamburg, who is looking at similar results and with whom I am now collaborating. |
| Sectors | Education |
| Description | LST(I) Investigations with Jonathan Osinski |
| Organisation | University of Hamburg |
| Department | Hamburg Observatory |
| Country | Germany |
| Sector | Academic/University |
| PI Contribution | I was recently invited to Hamburg to collaborate with Jonathan Osinski, further developing the outcomes of my EPSRC-funded research. Although this collaboration is very new, we already have what we believe is an answer to a long-standing question in the field, produced during an intensive discussion over the course of a couple of days. |
| Collaborator Contribution | The University of Hamburg (or more precisely, Benedikt Lowe, one of the professors there) arranged for me to fly to Hamburg and stay in a hotel for a week at their expense, in order to allow the collaboration to happen. Osinski and I worked out our main result over together in collaborative discussion. |
| Impact | We have discovered a new kind of result about LST numbers. (See research outcomes for an explanation of LST numbers). Previously, all that was really known about LST numbers was a lower bound and an upper bound. We have showed that the LST number can only possibly be a rare kind of number with certain special properties. We think that we have also shown that many of these special numbers are valid candidates for the LST number, so this appears to be a complete or near-complete account of the possible LST number values. |
| Start Year | 2023 |
| Description | Name principle investigations with Philipp Schlicht |
| Organisation | University of Bonn |
| Country | Germany |
| Sector | Academic/University |
| PI Contribution | My name principles work (described in other sections of this form) was done in collaboration with Schlicht, mostly over Zoom during lockdown. We contributed equally, both working on most of it together, bouncing ideas back and forth to the point that it is hard to separate out who wrote what, even within any particular sentence. However, the proof of what became our tentpole theorem was mine. |
| Collaborator Contribution | As I said above, we both worked on most of the work equally and most of our work cannot be separated. However, Schlicht was the one who originally introduced me to the ideas that turned into name principles. |
| Impact | A 50 page paper, accepted by the Annals of Pure and Applied Logic. |
| Start Year | 2019 |
| Description | Name principle investigations with Philipp Schlicht |
| Organisation | University of Bristol |
| Country | United Kingdom |
| Sector | Academic/University |
| PI Contribution | My name principles work (described in other sections of this form) was done in collaboration with Schlicht, mostly over Zoom during lockdown. We contributed equally, both working on most of it together, bouncing ideas back and forth to the point that it is hard to separate out who wrote what, even within any particular sentence. However, the proof of what became our tentpole theorem was mine. |
| Collaborator Contribution | As I said above, we both worked on most of the work equally and most of our work cannot be separated. However, Schlicht was the one who originally introduced me to the ideas that turned into name principles. |
| Impact | A 50 page paper, accepted by the Annals of Pure and Applied Logic. |
| Start Year | 2019 |
| Description | Name principle investigations with Philipp Schlicht |
| Organisation | University of Vienna |
| Department | Vienna Institute of Demography |
| Country | Austria |
| Sector | Academic/University |
| PI Contribution | My name principles work (described in other sections of this form) was done in collaboration with Schlicht, mostly over Zoom during lockdown. We contributed equally, both working on most of it together, bouncing ideas back and forth to the point that it is hard to separate out who wrote what, even within any particular sentence. However, the proof of what became our tentpole theorem was mine. |
| Collaborator Contribution | As I said above, we both worked on most of the work equally and most of our work cannot be separated. However, Schlicht was the one who originally introduced me to the ideas that turned into name principles. |
| Impact | A 50 page paper, accepted by the Annals of Pure and Applied Logic. |
| Start Year | 2019 |