The fourth meeting of LMS Set Theory in the UK network was be held at the University of Oxford on Saturday 14 December 2019. The speakers were Philipp Schlicht, Carolin Antos, John Howe, Joel David Hamkins, Philip Welch and Asaf Karagila.

Next Meeting: STUK5

The next meeting is at the Royal Society building in London, on the 11 February 2020. Here is the STUK5 website.

Venue & Time

We will meet at 10:30am in lecture room L4 at the Mathematical Institute. The following links provide general information about the building.

Schedule

10:30am Welcome refreshments
11:00am Philipp Schlicht: Internal absoluteness
12:00pm Lunch break — see lunch options
2:00pm Carolin Antos: Models in set-theoretic methodology:
A Second Philosophy account of the introduction of Forcing
3:00pm Refreshments
3:30pm John Howe: Ramsey Theory on trees and homogeneous structures
4:00pm Joel David Hamkins: Modal model theory
4:30pm Refreshments
5:00pm Philip Welch: From L[Card] to L[Reg]
5:30pm Asaf Karagila: one of the following, depending on audience preference
  • Having more things by forgetting how to count them
  • Iterated failures of choice
  • Collapsing cardinals without collapsing cardinals
  • What does preserving DC imply?
7:30pm Conference Dinner — see dinner information

Abstracts

Internal absoluteness

Philipp Schlicht, University of Bristol

Connections between generic absoluteness and regularity properties have been studied by Solovay, Bagaria, Brendle, Feng, Magidor, Woodin and others, but some central questions are open. We introduce a new viewpoint by working with principles of internal projective absoluteness between the universe and generic extensions of countable elementary submodels. The main result is an equivalence between such principles and modified regularity properties: for proper forcings of the form Borel/I, one has the equivalence of (1) internal absoluteness for Borel/I (2) 1-step projective absoluteness for Borel/I and the regularity property associated to I (3) the uniformization principle associated to I. Moreover, these principles are closely linked with universally Baire sets. This is joint work in progress with Sandra Müller.

Models in set-theoretic methodology: A Second Philosophy account of the introduction of Forcing

Carolin Antos, University of Konstanz

Penelope Maddy's account of naturalism in mathematics is extensive. In her 2011 book "Defending the Axioms" (DA), she applies this account - as spelled out in terms of her Second Philosophy - to set-theoretic methodology. Using four examples from set-theoretic practice, she analyzes the underlying methodology of the "evidential structure of the subject", carefully aiming at "the convictions that actually drive the practice". From this, she singles out set-theoretic methods that can then serve, for example, to defend certain axioms.

In this talk I want to apply Maddy's analysis to the set-theoretic practice of forcing. I will analyze the introduction of forcing by Cohen, proceeding as Maddy does in her case studies in DA. I will argue that the method that actually drives the practice here is a new type of proof method, namely a certain use of set-theoretic models. I will give an argument that this can indeed be considered as a “typical" practice in set theory and that the Second Philosophers account of set-theoretic methodology should be broadened to include this method. Finally, I will give an outline on how this broadened picture of set theory still complies with the foundational roles of set theory as recounted by Maddy in her "Set-theoretic Foundations” (2016).

Ramsey Theory on trees and homogeneous structures

John Howe, University of Leeds

Milliken's theorem on infinitely branching trees, and variants of it, have found various applications in the Ramsey theory of homogeneous structures. I will endeavour to explain this connection, discussing the classical case of the rationals, and hopefully some recent work on permutation structures.

Modal model theory

Joel David Hamkins, University of Oxford

I shall introduce the subject of modal model theory, a research effort bringing modal concepts and vocabulary into model theory. For any first-order theory T, we may naturally consider the models of T as a Kripke model under the submodel relation, and thereby naturally expand the language of T to include the modal operators. In the class of all graphs, for example, a statement is possible in a graph, if it is true in some larger graph, having that graph as an induced subgraph, and a statement is necessary when it is true in all such larger graphs. The modal expansion of the language is quite powerful: in graphs it can express k-colorability and even finiteness and countability. The main idea applies to any collection of models with an extension concept. The principal questions are: what are the modal validities exhibited by the class of models or by individual models? For example, a countable graph validates S5 for graph theoretic assertions with parameters, for example, just in case it is the countable random graph; and without parameters, just in case it is universal for all finite graphs. Similar results apply with digraphs, groups, fields and orders. This is joint work with Wojciech Wołoszyn.

From L[Card] to L[Reg]

Philip Welch, University of Bristol

We show how to extend previous results characterising L[Card] as a generalised Prikry extension of its core L[E]-model, to models with predicates for the Inaccessible cardinals.

Having more things by forgetting how to count them

Asaf Karagila, University of East Anglia

In this joint work with Philipp Schlicht we show that forcing over Cohen's model to well-order the canonical Dedekind-finite set we actually get a generic extension where the new enumeration is generic (over the original ground model) for adding that-many Cohen reals to begin with. In particular, that means that by forgetting how to count the set of Cohen reals we can re-organise it to whatever size we desire.

Iterated failures of choice

Asaf Karagila, University of East Anglia

We provide an example of iterating symmetric extensions where each step we add a "properly new set" without adding sets of sufficiently small rank to the universe. We then show how to utilise this to prove, for example, that every partial order can be realised by cardinals in the same model of ZF.

Collapsing cardinals without collapsing cardinals

Asaf Karagila, University of East Anglia

In this joint work with Philipp Schlicht we show that one can, in some sense, force with a collapse forcing over Cohen's first model and add a surjection from a Dedekind-finite set onto an arbitrarily large ordinal, without adding any new sets of ordinals. In particular, no cardinals will be collapsed.

What does preserving DC imply?

Asaf Karagila, University of East Anglia

We know that if a symmetric system is given, and the forcing satisfy certain closure or chain conditions, and the filter of subgroups is sufficiently close, Dependent Choice will be preserved when passing to the symmetric extension. But what about the converse? We will discuss what is known and not known, and how could one attack this open problem.

Lunch Options

There is no prearranged lunch plan. Attendees are free to make their own arrangements during the two hour lunch break 12:00pm–2:00pm. We can offer the following suggestions.

  • Little Clarendon Street is 5 minutes walk away from the Mathematical Institute, and offers an number of nice places to grab lunch, from sandwiches and bagels to Lebanese cuisine. Here is Little Clarendon Street on a map.
  • Walton Street, also just around the corner, has a large selection of sit-down cafés and restaurants. Here is Walton Street on a map.
  • There are two Co-op supermarkets located on Walton Street. Here is a map of the closest Co-op.
  • There is also a large Tesco supermarket 10 minutes walk away. Here is Tescos on a map.
  • For something a little different, consider the Ashmolean Museum of art and archaeology. This fantastic free museum is 10 minutes walk away, and houses a vast collection of objects. It boasts a rooftop restaurant, as well as a smaller café on the lower level. Here is The Ashmolean Museum on a map.
  • Please get in contact with us via the email addresses below if you have any further questions, for example concerning accessibility.

Conference Dinner

A conference dinner will be held at 7:30pm in The Crown on Cornmarket Street, which is a 10–15 minute walk away from the Mathematical Institute. Here is The Crown on a map. Here is the address:

The Crown
59a Cornmarket Street
Oxford
OX1 3HB

Contact

If you have any queries about the meeting, please contact Sam Adam-Day at samuel.adam-day@maths.ox.ac.uk or Joel David Hamkins at joeldavid.hamkins@philosophy.ox.ac.uk.

Previous Meetings

  • STUK 0: 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

Next Meeting

STUK 5: the Royal Society, London, 11 February 2020