Postscript: The Impact of Mathematical Logic

Abstract

In all of the previous chapters we have dealt with developments which started before the middle of the Twentieth Century, although, in many cases, we also reported on some of their later phases. In the present chapter we are making an exception to this rule which we imposed on our account in order to keep it from becoming too voluminous. Our reason for doing so is the exceptional nature of the impact of mathematical logic on combinatorial group theory. We encounter here the situation where concepts and methods from one mathematical discipline become essential ingredients of the theorems and of the answers to previously open questions of another discipline. Of course, events of this type have happened before, but they deserve to be mentioned with distinction. In a way, the connection between topology and combinatorial group theory provides a similar example. However, there exists a notable difference between these two occurrences. Topology and combinatorial group theory grew up together, but the event we are now going to describe involves the action of a highly developed sophisticated theory on another equally developed discipline.