of East Anglia, England: Combinatorics of trees.
In the recent years I have been working on various aspects of the
combinatorics of trees. The idea has been to consider trees of size
κ which do not have a branch of size κ, for various
κ. This class can be endowed with a notion of embedding, where one
tree embeds into another if there is a strict order preserving function
between them. The function does not have to be injective. These concepts
were highlighted and studied to a great depth by those studying infinitary
logics, notably Vää,nänen. Many structural theorems are now
especially about the situation when κ is ω1. I
discuss some of these results, especially the results involving the
behaviour of the class with respect to universality, where my research has
mostly concentrated. I shall also discuss the rather unexpected behaviour
when κ is a singular cardinal.
Universität Wien, Austria: Consistency completeness.
As ZFC is incomplete, there are statements which,
though not provable in ZFC, are nevertheless consistent
But some statements are more consistent than
others. We regard the universe V as a
countable transitive ZFC model and use the term
proper class model of ZFC
to refer to transitive ZFC models of the same
ordinal height as V. If M and N are proper class
is an inner model of N iff it is a submodel of N and
is an outer
model of N iff it contains N as a submodel. The models
M and N
are compatible iff they have a common outer model.
A statement is
- consistent with the ordinals iff it holds in some
proper class model,
- consistent with V
iff it holds in some model compatible with V,
- internally consistent iff it holds in some inner
model of V,
- externally consistent iff it
holds in some outer model of V.
The model V is
- complete for consistency iff any statement consistent with
is consistent with all outer models of V,
- complete for internal consistency iff any statement true in an
inner model of some outer model of V is already true in an inner
model of V,
- complete for external consistency iff any statement true in
outer model of V is true in an outer model of any outer model of
Proposition. If V is complete for internal consistency or
external consistency then it is
also complete for consistency.
Theorem. Assume the consistency of a Woodin
cardinal and an inaccessible above. Then there is a countable
transitive model V which is complete for both internal and
There is also a version of completeness for internal consistency,
called the Strong Inner Model Hypothesis (SIMH), which
introduces "absolute parameters".
It is not known if the SIMH is consistent. However we do have the
following partial result.
SIMH for the parameter ω1 is consistent, relative to a
Woodin cardinal and an inaccessible above.
Dualities between algebras and topological spaces play a central role
relating syntactic and
semantic approaches to propositional and geometric logics of various
kinds, and highly sophisticated
tools that fit into this framework are available in focused areas of
research (such as modal logic).
Recent advances in the study of duality theory in the guise of lattice
theoretic completions has allowed the translation of these tools to a much
wider setting including that of substructural logic and that of the model
theory of finite algebras.
This talk elaborates on this point of view and surveys a few recent
Palmgren, Uppsala Universitet, Sweden: Point-free topology
versus topology according to Brouwer and
Locale theory and formal topology are two essentially equivalent ways
studying spaces in terms of their complete lattice of open sets rather
than in terms of their points. From a classical set-theoretic perspective,
there is a straightforward relation between the point-free and point-based
approach for important classes of spaces of interest in analysis. This
situation changes rather drastically when considering them from a
constructive and predicative point of view, which we do in this talk. The
relation of formal topology to the traditional constructive versions of
topology -that of Brouwer which uses non-recursive axioms like the FAN
theorem, and that of Bishop which is largely confined to separable metric
spaces- is not completely understood. It is well-known that point-free
topology can prove many results which follows from FAN, such as the
Heine-Borel theorem, provided they expressed using inductively generated
open covers. In this talk we present a full and faithful embedding of the
category of locally compact metric spaces (in Bishop's sense) into the
category of formal spaces, using Steve Vickers' localic completion method.
Properties of this embedding shed some light on Bishop's definition of a
continuous map on an open subset. They also show how some of the
difficulties of Bishop's approach, such as the problem to get a good
category of spaces which includes the reciprocal map, can be resolved in
the setting of point-free topology. Transfer principles for inequalities
on the different categories of spaces are given.
Università di Padova, Italy: Minimalist foundation and
pluralism in mathematics:
computation and structure in topology.
Abandoning the classical view, one is faced with a plurality of
apparently incompatible proposals for the foundation of constructive
mathematics. We have shown that actually there is a common base, of which
other foundations can be seen as extensions, and which thus has been
called the minimalist foundation (joint work with M. E. Maietti). Its
positive feature is to allow a formal treatment both for the computational
and the structural aspects of mathematics; this is obtained by
distinguishing within a single formal system two different but connected
levels of abstraction.
When used to develop topology with no subjection to the definitions
reached in the classical approach, the minimalist foundation brings to the
discovery of some simple new algebraic structures. Their peculiarity is
the presence of primitives for existential notions, mainly overlap and
positivity, which are dual to the standard ones, that is inclusion and
cover. One can show that this extra expressive power is sufficient for a
purely algebraic treatment of the main concepts of topology. All starts
from the discovery that, when stated in an intuitionistic and effective
way, the notions of open and closed are perfectly dual to each other. In
general, one can realize that topology and logic are much more intertwined
than one would expect before.
Using the minimalist foundation, one can perceive the role of different
foundational assumptions in the practice of mathematics. In particular,
one can realize that sometimes stronger principles obscure the perception
of some natural mathematical structures. The moral is that each different
foundation brings to the development of a different kind of mathematics
and hence that pluralism is a source of richness.
Truss proved that the automorphism group of the random
graph is a simple group. In joint work with Dugald
Macpherson we generalize this to relational structures
with free amalgamation.
Starting from the late 1980s Shelah developed
PCF theory and used it to obtain striking results in Cardinal
A celebrated theorem of Shelah states that if
strong limit cardinal then
This is achieved by definying a topological space, called the PCF space,
with certain combinatorial properties and showing that any such space
have size less than ω4.
The best lower bound on
the other hand
One might hope to improve Shelah's upper bound
by showing that the any PCF space is of size less than
We relate this problem to the study of thin tall locally compact
spaces and show that it is possible to have PCF spaces of height any
ordinal less than ω3, thus Shelah's upper bound