A one-day conference in the series *Set theory and its neighbours*
will take place on Wednesday 5th December
at the Department of Mathematics, University
College London, 25 Gordon Street, London, WC1.

The talks will be in room 706, with the
first talk starting at **2pm**. (Note the later starting time than
usual.)
There will be tea (in room 606) from 4-4.30, and the meeting
should finish by approximately 6.40pm.

The speakers at the meeting will be:

- Andrew Brooke-Taylor (Bristol)

*Large Cardinals and Definable Well-Orderings of the Universe*

**Abstract**: One very nice property of Goedel's constructible universe*L*is that it carries a definable well-ordering of all sets. Of course, many large cardinals axioms are incompatible with the assumption that*V=L*, but nevertheless we can still ask whether they are compatible with the existence of a definable well-ordering of the universe. We shall exhibit a forcing construction that demonstrates for many large cardinal axioms that they are indeed compatible, using a coding scheme involving the principle ◊^{*}.

- Lylah Haynes (Birmingham)

*MCP, MδN and factorizations of MN*

**Abstract**: Ishikawa characterized a countably paracompact space as one in which given any decreasing sequence of closed sets*(D*with empty intersection, there exists a sequence of open sets_{i})*(U*whose closures have empty intersection such that_{i})*U*contains_{i}*D*for each_{i}*i*. A monotone version of Ishikawa's property, known as monotone countable paracompactness (MCP), was introduced by Good, Knight and Stares who carried out an extensive survey of it. In many cases, the set-theoretic assumptions necessary for results concerning countable paracompactness may be abandoned with MCP. In this way MCP is analogous to monotone normality; set-theoretic assumptions may be dropped when replacing 'normal' with 'monotonically normal' in certain results.The study of MCP inspired us to investigate monotone versions of different characterizations of countable paracompactness. In particular, monotonizing Mack's result that a space is countably paracompact iff its product with [0,1] is δ-normal led us to the notion of monotone δ-normality (MδN). It turns out that if the product of a space

*X*with [0, 1] is MδN then*X*is MCP, however the converse is not true.Since much is known about MCP, it seems natural to ask about the behaviour of MδN and other monotone versions of δ-normality. We show how several monotonizations of δ-normality interrelate and provide factorizations of monotone normality in terms of these properties.

- Achim Jung (Birmingham)

*From valuations to measures*

**Abstract**: There is a long history of the use of topology in computer science, going back at least to the work of Kleene in the 50s. The structures considered in*Denotational Semantics*, in particular, are best understood as certain topological spaces. Unfortunately, however, the spaces are just T_{0}and not Hausdorff and so many of our ingrained topological intuitions do not apply.I will begin the talk with describing the particular breed of T

_{0}space that tends to arise in semantics, called*stably compact space*, and I will explain that it relates one-to-one to the much better known concept of a*compact ordered space*.When trying to define

*Borel measure*on a stably compact space, one finds that it is convenient to begin with the much simpler concept of a*valuation*, which provides a "volume" for the open sets only. I will describe these and show that there is a perfectly good notion of integration for (semi) continuous functions with respect to a valuation.The question then arises whether a valuation on a stably compact space will determine a Borel measure on the associated compact ordered space. This is indeed the case and the remainder of the talk will be devoted to a particularly elegant argument for showing this. (It was my co-author Klaus Keimel who first suggested this approach.)

- Balazs Montagh (UCL)

*Ramsey's Theorem for graphs under degree conditions*

**Abstract**: tba

- Brian Semmes (Amsterdam)

*Decomposing 2 → 3 and 3 → 3*

**Abstract**: In this talk, I will review game-theoretic characterizations of the continuous, Baire class 1, and Baire class 2 functions, and describe how the Jayne-Rogers theorem can be generalized using game-theoretic methods. Let "*n → m*" denote the class of functions for which the preimage of a boldface**Σ**^{0}_{n}set is boldface**Σ**^{0}_{m}. The Jayne-Rogers theorem states (with very minimal assumptions) that the 2 → 2 functions are piecewise continuous. The results I describe in this talk are for the 2 → 3 and 3 → 3 classes, for functions on the Baire space.

We aim to keep the meetings fairly relaxed, allowing plenty of opportunity for informal discussion. We welcome and encourage anyone to participate. Please do tell anyone about the meeting who you think may be interested in it. There is no registration fee for the meeting. We are happy for you to email us to let us know if you intend to come, but you are also very welcome simply to turn up on the day if you make a late decision. And let us know if you would like to speak or have ideas for speakers at future meetings. After the meeting we will probably go for a near-by drink and then supper.

Map of UCL and environs. 25 Gordon Square is at the top right hand corner of square E2.
Return to the
*
Set theory and its neighbours* homepage
for information, including slides
from the talks and related preprints, about the previous meetings.

Last updated on 15th November 2007, Charles Morgan