This page gives information about one-day conference
on **Finite model theory and set theory**, the second
in the series *Set theory and its neighbours*,
which took place on
**Thursday 7th January 1999** at the London
Mathematical Society building, De Morgan House, 57-58 Russell
Square, London WC1.

The timetable for the meeting was as follows.

- 11.00: Coffee
- 11.30:
Anuj Dawar (Cambridge, Britain)
*"Fixed point logics on finite and infinite stuctures"*(ps) - 12.30: Arthur Apter
(CUNY, USA)
*"Strong compactness and Laver indestructibility"*(LaTeX, dvi or ps) - 1.20: Lunch
- 2.10:
Péter Komjáth (Eötvös, Hungary)
*"Triple systems with uncountable chromatic number"* - 3.00:
Adrian Mathias (Universidad de los Andes, Columbia)
*A totally unscripted talk on AD* - 3.50:
Philip Welch (Kobe, Japan)
*"The length of infinite time Turing machine computations"*(LaTeX, dvi or ps) - 4.40: Tea
- 5.10:
James Cummings (Carnegie Mellon, USA)
*"Scales, squares and stationarity"*(LaTeX, dvi or ps)

- James Cummings and
Matt Foreman,

*"Condensation coherent squares and mutually stationary sets"*(ps) - 6.00:
Martin Otto (Aachen, Germany)
*"Descriptive complexity and semantically defined fragments of Ptime"*(Abstract) (Slides, ps) - 7.00: Adjurn to Cafe/Bar

**Why FMT and ST?**
In the series we eschew consideration of superficial applications of
one subject to another, aiming instead to focus on intrinsic
relationships, both concrete and thematic. Finite model theory and
set theory are a priori distinct, the former dealing with finite and
the latter with infinite structures. However beyond this one is
immediately struck by the basic role of abstract (as opposed to
applied) model theory and by the importance of extensions of first
order logic in each. As Otto says in his recent Springer monograph on
finite model theory, ``first order logic lacks recursion and
counting''[...=ordinals and cardinals].) And one also sees that
both are vitally concerned with combinatorial questions and in
particular with what can be done in a subexponential number of steps.

At this meeting the specifically finite model theory components addressed questions in descriptive complexity about logics capturing complexity classes (with less emphasis on 0-1 laws, the other main strand of finite model theory). Here the central problem is understanding canonisation with respect to different logics, i.e., classifying salient equivalence relations -- the very theme of Farah and Hjorth's talks at the first meeting (on Set theory, analysis and their neighbours). One has the same problems in canonisation as in classifying equivalence relations in combinatorial set theory. Essentially the principal technique available to enable one to reduce one equivalence relation to another effectively is bare-handed combinatorics, and the lack of a suitable analogue of the Baire Category Theorem (for similar reasons in both cases) has hitherto frustrated progress on distinguishing the complexity of relations.

In view of these strong links we think that a combined meeting on finite model theory and set theory wass intellectually coherent. We attracted a wide range of logicians and other mathematicians from the UK (and further afield), and we hope to encourage communication and interaction between people working in these closely related fields and their neighbours.

We are very grateful to the British Logic Colloquium for helping to provide financial support for the meeting. We are also grateful to the LMS for allowing us to use De Morgan House as a venue.

*Topology and set theory* (the fiftth meeting in the series),

*Ergodic theory and set theory* (the fourth meeting in the series),
including
slides and abstracts from the talks.

*Combinatorics
and set theory* (the third meeting in the series), including
slides from the
talks and related preprints.

*Set theory,
analysis and their neighbours* (the first meeting in the series),
inlcuding
slides from the talks and related preprints.

Last updated on 4th February 1999