# STUK 7:

*Still STUK @ Home*

### World Logic Day

14 January 2022

online

The seventh meeting of the *Set Theory in the UK* network
will be held online on
**Friday the 14th of January, 2022,**
as part of
World
Logic Day.
All are welcome to attend. Speakers will include

- Jonathan Schilhan:
*Sequential and distributive forcing without choice* - Šárka Stejskalová:
*The negation of the Weak Kurepa Hypothesis* - John Truss:
*Dedekind-finite Cardinals having countable partitions*

In the afternoon we will have a series of informal minitalks, particularly encouraging local graduate students to introduce themselves to the STUK community if they are new, and to talk about what they are working on.

At the end of the conference we will have a social on Gather.Town, which is a virtual room, allowing you to walk around as an avatar and talk to the other participants. Please join us for this so that we can all meet each other, albeit virtually.

### Coordinates and Contact

The meeting will happen via Zoom; on the day, click here to attend the talks (meeting ID: 856 0070 6394). The password is formed as follows: take the 10 letter word, starting with "F" and with all other letters lower-case, which is the name of the ZFC axiom which guarantees in particular that no set can be an element of itself. Then, replace each "o" by "0" and each "a" by "@", and you have the password.

Click here for the social session at the end of the day. The password is the same as for the Zoom session.

The organisers are Richard Matthews (r.m.a.matthews@ leeds.ac.uk) and Andrew Brooke-Taylor (a.d.brooke-taylor@ leeds.ac.uk). We particularly encourage you to get in touch if you would like to volunteer a mini-talk.

This meeting is organised from the University of Leeds. Previous iterations of the STUK meetings were supported by the LMS.

### Timetable

All times are UK (GMT).10:00 | Initial tea & chat; Welcome |

10:15 | Jonathan Schilhan (slides) |

11:15 | Tea break |

11:45 | John Truss (slides) |

12:45 | Lunch |

2:00 | Šárka Stejskalová (slides) |

3:00 | Tea break |

3:30 | Introductions Minitalks: Bea Adam-Day (slides) Tanmay Inamdar (slides) Paul Levy (slides) |

4:45 | Gathertown social |

### Talk titles and abstracts

**Jonathan Schilhan,** University of East Anglia.

*Sequential and distributive forcing without choice.*

Forcing over models of ZFC has a long history and is quite well understood.
But many basic results rely strongly on the Axiom of Choice
and so naturally much less is known if we only assume ZF.
One such result is the equivalence of distributivity and sequentiality
(not adding sequences of certain length).
Interestingly, this equivalence can break down without choice.
Another central topic is the preservation of weak choice principles such as DC
(dependent choice). It is known that sigma-closed forcing preserves DC.
So what about sequential or distributive forcing?

**John Truss**, University of Leeds.

*Dedekind-finite Cardinals having countable partitions*

We study the possible structures which can be carried by sets which have no countable subset,
but which fail to be `surjectively Dedekind finite', in two possible
senses, that there is a surjection to omega, or alternatively, that there is a surjection to a proper superset.

**Šárka Stejskalová,** Department of Logic, Charles University and
Institute of Mathematics, Czech Academy of Sciences.

*The negation of the Weak Kurepa Hypothesis.*

Abstract (pdf)

#### Minitalks:

**Bea Adam-Day,** University of Leeds.

*C(n) large cardinals*

**Tanmay Inamdar,** Bar-Ilan University.

*When Sierpinski met Ulam*

Abstract (pdf)

**Paul Levy,** University of Birmingham.

*Broad infinity and Generation Principles*

We introduce Broad Infinity, a new set-theoretic axiom scheme that (in my opinion) is intuitively plausible. It states that "broad numbers", which are three-dimensional trees whose growth is controlled by a specified class function, form a set. The Broad Infinity scheme is equivalent, assuming the Axiom of Choice, to the widely studied Ord-is-Mahlo scheme: every closed unbounded class of ordinals contains a regular ordinal.
Whereas the axiom of Infinity leads to generation principles for sets and families and ordinals, Broad Infinity leads to more advanced versions of these principles. We relate these principles under various prior assumptions: the Axiom of Choice, the Law of Excluded Middle, and weaker assumptions.

### Previous meetings

- STUK 0 (preliminary meeting): The Royal Society, London, 7 November 2018
- STUK 1: University of Cambridge, 16 February 2019
- STUK 2: University of Bristol, 8 May 2019
- STUK 3: University of Leeds, 3 October 2019
- STUK 4: University of Oxford, 14 December 2019
- STUK 5: The Royal Society, London, 11 February 2020
- STUK 6
*(STUK @ Home)*: Online, 4 December 2020