Juha Oikkonen

Abelian p-groups with no invariants☆

Abstract We prove for abelian p -groups a non-structure theorem relative to approximations of Ehrenfeucht-Fraisse games of length ω 1 in terms of linear orderings with no uncountable descending sequences. Our result shows that there is a group which is too complicated to be characterized up to isomorphism by the Ehrenfeucht-Fraisse game approximated by a fixed ordering. This means that such a group cannot have any complete invariants which are bounded in the sense of these approximations of the Ehrenfeucht-Fraisse game. On the other hand, all the approximations characterize together the notion of isomorphism among groups of cardinality at most ω 1 . From the point of view of Stability Theory, our result concerns certain stable theories with NDOP and NOTOP.

Some reasons for change in undergraduate mathematics enrolments

Here, we look at the enrolments of students in undergraduate mathematics courses in a number of countries. The data show various increases and decreases and we suggest some common reasons for the fluctuations. These include students’ goals of a secure and well-paid job, government actions and the state of the economy in the country concerned. We consider several ways in which departments have successfully approached downturns in numbers by their interactions with students by introducing new teaching approaches, using technology and establishing mathematics centres.

Ideas and results in teaching beginning maths students

At the University of Helsinki, special effort has been put to help beginning maths students to get started with their studies. This involves new ways of teaching a lecture course and new ways of giving peer support to the students. The results have been promising: the number of students passing certain important courses has been at least doubled compared to the earlier prevailing level. The present article describes the work done in connection to a lecture course and the thinking that lies behind it. A paper on peer support is planned.

Logic Colloquium '90

This volume is the Proceedings of Logic Colloquium 90 held in Helsinki, Finland, from July 15 to July 22, 1990. The congress belongs to the annual series of the European Summer Meetings of the Association for Symbolic Logic. These proceedings contain 13 research papers written mainly by the invited speakers of the congress. They represent all fields of mathematical logic from the philosophy of mathematics through model theory, proof theory, recursion theory and set theory, to connections of logic to computer science.

Game-theoretic inductive definability

Abstract Oikkonen, J. and J. Vaananen, Game-theoretic inductive definability, Annals of Pure and Applied Logic 65 (1993) 265-306. We use game-theoretic ideas to define a generalization of the notion of inductive definability. This approach allows induction along non-well-founded trees. Our definition depends on an underlying partial ordering of the objects. In this ordering every countable ascending sequence is assumed to have a unique supremum which enables us to go over limits. We establish basic properties of this induction and examine examples where it emerges naturally. In the main results we prove an abstract Kleene Theorem and restricted versions of the Stage-Comparison Theorem and the Reduction Theorem.

Mathematics Education at Tertiary Level and Access to Tertiary Level

Co-chairs: Ansie Harding (South Africa), Juha Oikkonen (Finland); Team Members: Christopher Sangwin (UK), Sepideh Stewart (New Zealand), Miroslav Lovric (Canada), Sung-Ock Kim (Korea); Liaison IPC Member: Johann Engelbrecht (South Africa).

Hierarchies of Model Theoretic Definability—an Approach to Second Order Logics

We define hierarchies of model theoretic definability by iterating PC- and RPC-definitions and their duals. These hierarchies can be considered generalized second order logics. We show that the model theoretic RPC-hierarchy defined in terms of L \(\infty\) \(\omega\) (WF) essentially coincides with the Levy hierarchy of classes. These results can be applied to get new information about L 11.

A Generalization of the Infinitely Deep Languages of Hintikka and Rantala

Jaakko Hintikka and Veikko Rantala define in [2] infinitary languages in which sentences may be of infinite depth. Recall that the syntactic structure of the sentences of ordinary finitary as well as infinitary logic may be described in terms of certain trees.