Preface

Summary

The subject matter of this book constitutes a merging of several directions of work in general model theory over the last 25 years. Three main lines can be distinguished: first, that initiated by Andrzej Mostowski on cardinality quantifiers in the late 1950s; second, the work of Alfred Tarski, his colleagues and students on infinitary languages into the mid-1960s; and, finally, that stemming from the results of Per Lindström on generalized quantifiers and abstract characterizations of first-order logic in the late 1960s. The subject ofabstract model theory blossomed from that as a unified and illuminating framework in which to organize, compare, and seek out the properties of the many stronger logics which had then come to be recognized.

Interest in abstract model theory and extended logics was intense in the early 1970s, particularly as a result of the work of Jon Barwise on infinitary admissible languages. Where the previous developments had largely connected up model theory with set theory, this added ideas from extended recursion theory in an essential way, e.g., to yield successful infinitary generalizations of the compactness theorem. It also turned out that proof theory—including such consequences as the interpolation and definability theorems—could be successfully generalized to these languages. Thus, one was here witness to an exciting confluence of all the main branches of mathematical logic. These successes led to promising research programs for further interactive development, but the hopes they raised, especially with respect to the treatment of uncountable languages, were not realized. A number of us (including Barwise and myself) who had been involved at that stage of the subject turned to other interests in the latter part of the 1970s and gave little attention to its ongoing progress. As it happens, all through that period (at least) the set-theoretic and model-theoretic aspects of the subject were continuing to develop at a rapid rate. Looking again at the field in 1980 we found a body of work that was quite staggering.

The re-examination of the area at that time had come in response to repeated urging by the editors of the Q-group (particularly by Gert Muller) for Barwise and/or me to write a volume for the series on the general subject of model-theoretic logics.