E W Beth Centenary
Minisymposium on Mathematical Logic

Abstracts.

Mirna Dzamonja, University 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 known, especially about the situation when κ is ω₁. I will 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.

Sy Friedman, Universität Wien, Austria: Consistency completeness.

As ZFC is incomplete, there are statements which, though not provable in ZFC, are nevertheless consistent with ZFC. 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 models then M 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 V 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 an outer model of V is true in an outer model of any outer model of V.

Proposition. If V is complete for internal consistency or for 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 external consistency.

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.

Theorem. 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 in 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 results.

Erik Palmgren, Uppsala Universitet, Sweden: Point-free topology versus topology according to Brouwer and Bishop.

Locale theory and formal topology are two essentially equivalent ways of 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.

Giovanni Sambin, 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 Arithmetic. A celebrated theorem of Shelah states that if אּ_ω is a strong limit cardinal then 2^אּ_ω < אּ_ω4. This is achieved by definying a topological space, called the PCF space, with certain combinatorial properties and showing that any such space must have size less than ω₄. The best lower bound on the other hand is אּ_ω1. One might hope to improve Shelah's upper bound by showing that the any PCF space is of size less than אּ₃, אּ₂, or even אּ₁. We relate this problem to the study of thin tall locally compact scattered spaces and show that it is possible to have PCF spaces of height any ordinal less than ω₃, thus Shelah's upper bound cannot be significantly improved.