Summary. K Gödel published two seminal papers on general relativity theory and its application to the study of cosmology. The first examined a non-expanding but rotating solution of the Einstein field equations, in which causality is violated; this lead to an in-depth examination of the concepts of causality and time in curved space-times. The second examined properties of a family of rotating and expanding spatially homogeneous solutions of the Einstein equations, which was a forerunner of many studies of such cosmologies. Together they stimulated examination of themes that were fundamental in the development of the Hawking-Penrose singularity theorems and in studies of cosmological dynamics. I review these two papers, and the developments that resulted from them.

Introduction

Gödel became interested in general relativity theory while he and Einstein were both on staff of the Institute for Advanced Studies in Princeton. Apparently they discussed the subject together often. His resultant two papers had a major impact:

On Perspectives. Mathematical logic arose from a concern with the nature and the limits of rational or mathematical thought, and from a desire to systematise the modes of its expression. The pioneering investigations were diverse and largely autonomous. As time passed, and more particularly in the last two decades, interconnections between different lines of research and links with other branches of mathematics proliferated. The subject is now both rich and varied. It is the aim of the series to provide, as it were, maps of guides to this complex terrain. We shall not aim at encyclopaedic coverage; nor do we wish to prescribe, like Euclid, a definitive version of the elements of the subject. We are not committed to any particular philosophical programme. Nevertheless we have tried by critical discussion to ensure that each book represents a coherent line of thought; and that, by developing certain themes, it will be of greater interest than a mere assemblage of results and techniques.

The books in the series differ in level: some are introductory, some highly specialised. They also differ in scope: some offer a wide view of an area, others present a single line of thought. Each book is, at its own level, reasonably self-contained. Although no book depends on another as prerequisite, we have encouraged authors to fit their book in with other planned volumes, sometimes deliberately seeking coverage of the same material from different points of view. We have tried to attain a reasonable degree of uniformity of notation and arrangement. However, the books in the series are written by individual authors, not by the group. Plans for books are discussed and argued about at length. Later, encouragement is given and revisions suggested. But it is the authors who do the work; if, as we hope, the series proves of value, the credit will be theirs.

The 1990 European Summer Meeting of the Association for Symbolic Logic was held in Finland from July 15 to July 22, 1990. The meeting was called Logic Colloquium '90 and it took place in the Porthania building of the University of Helsinki as part of the program of the 350th anniversary of the university.

The meeting was attended by 140 registered participants and 31 accompanying persons, from 23 different countries. The organizing bodies were the Department of Mathematics of the University of Helsinki, The Philosophical Society of Finland, and The Finnish Mathematical Society. Financial support was received from the Ministry of Education of Finland, The Academy of Finland, Suomen Kulttuurirahasto Foundation, UNESCO, Rolf Nevanlinna Institute, and IUHPS.

The Organizing Committee of the meeting consisted of Aapo Halko, Heikki Heikkila, Lauri Hella, Taneli Huuskonen, Tapani Hyttinen, Kerkko Luosto, Ilkka Niiniluoto (vice chairman), Juha Oikkonen, and Jouko Vaananen (chairman), all from the University of Helsinki.

The Program Committee consisted of Peter Aczel (Manchester), Max Dickmann (Paris), Heinz-Dieter Ebbinghaus (Freiburg), Jens Fenstad (Oslo), Jaakko Hintikka (chairman, Boston), Wilfrid Hodges (London), Alistair Lachlan (Vancouver), Azriel Levy (Jerusalem), Heikki Mannila (Helsinki), Ilkka Niiniluoto (Helsinki), Juha Oikkonen (Helsinki), and Jouko Vaananen (secretary, Helsinki).

The program of the meeting is listed on the following pages. Warren Goldfarb, Ronald Jensen, Phokion Kolaitis, Per Martin-Lδf, Alan Mekler, and Hugh Woodin did not contribute a paper to the proceedings. As comparison between the contents of this book and the actual program reveals, some authors made an agreement with the editors to contribute a slightly different paper from the one read in the meeting. Also Joan Moschovakis and Alan Silver were approached by the editors and they submitted the paper they read in a contributed papers session of the meeting.

The editors are indebted to Heikki Heikkila, Yiannis Moschovakis, Martti Nikunen, and Hannele Salminen for their help during the preparation of this volume. We owe special thanks to Herbert Enderton for the substantial work he has done in putting together the final manuscript.

In 1949 Gödel presented Einstein with a new solution of the field equations of the general theory of relativity: an exact solution which allows for the bizarre possibility of time-travel. This discovery of a universe with a strange time-structure which is consistent with the general theory of relativity did not however surprise Einstein, or so at least he leads the reader to believe. In his reply to Gödel's contribution, Einstein intimated that the strange feature of a possible closed time-like lines in which the distinction “earlier-later” should be abandoned, “disturbed… (him) already at the time of the building up of the general theory of relativity, without … [him) having succeeded in clarifying it.” The weird result of Gödel constituted in Einstein's view “an important contribution to the general theory of relativity, especially to the analysis of the concept of time.”

Unlike the general theory, the special theory of relativity has been accepted by the mid-century as an undisputed part of theoretical physics. It has been commonly presented as the invariance theory of the Maxwell equations under the Lorentz group of transformations. However, Einstein seems to have conceived of the special theory as the theory of a special type of gravitational field: the uniform one. This recognition motivated him to go further and require generalization by the inclusion of gravitational phenomena. In this move one wishes that the “causal” structure of space-time in general relativity should manifest locally the same qualities as the flat space-time of special relativity, though globally significant differences are expected to emerge.