Andréka, H. Madarász, J. Németi, I. Sági, G. and Sain, I.

Analyzing the logical structure of relativity theory via model theoretic logic. (Preliminary draft version, comments are welcome.)

This is a lecture notes for a course given March 1 - April 20, 1998 in CCSOM of the University of Amsterdam. The course description included below serves as an abstract.

Course description

This course is mainly designed for logicians and people having a strong logical background who are interested in using logic for gaining a deeper understanding of (at least part of) reality. This might include e.g. people studying the methodology or philosophy of sciences.

Historical perspective. Tarski formalized geometry as a theory of first order logic. The point here is to use only first order logic; no external ``devices'' or tacit assumptions are allowed to enter the picture. Motivated by Tarski, P. Suppes raised the problem of formalizing the theory of special relativity as a theory purely in first order logic. Among other things, we will do this, and then push ahead towards further goals (we will also briefly discuss works of Ax, Goldblatt and others in this direction). Our version of relativity theory will consist of a small number of axioms which are not only in first order logic, but are moreover easily comprehensible.

Logical core. In a few papers, Johan van Benthem elaborates the idea of separating out the ``logical cores'' of certain logics. The idea here is separating out the really essential part (from the logical point of view) from the whole ``burden of mathematical machinery attached to the subject in the course of time''. In the same ``Benthemian'' spirit we will hereby clearing the way for asking logical questions, concerning e.g. the logical structure of the theory, the number of non-elementarily-equivalent models, classification of models, etc. Among other things, we will use logic to find out which axioms are responsible for certain surprising predictions of relativity theory like e.g. ``no observer can move faster than the speed of light'', ``the twin paradox'' or issues concerning the (im)possibility of time travel.

Searching for insight. All discussions will be in terms of simple concepts. When formalizing our (language and) axioms we will confine ourselves to a very plain language, using such understandable expressions as ``bodies'' or ``observers''. Whenever we need more complex expressions like ``energy'', ``enthropy'' or ``curvature of space-time'', then we will first define these, as a logician would do, in terms of our plain language. This way we hope to gain real insight into why certain exotic predictions of relativity theory are ``predicted''. It also allows us to make the axioms with which we started subject to debate: both because of the plain language in which they are expressed and because of the purely logical nature of our reasoning.

Future perspectives. Hawking, Weinberg and others suggested the possibility of a final Theory of Everything. In the literature it is often argued that Gödel's incompleteness theorem renders such theory impossible. This is a challenge for the logician. After the course, we will be in a better position for progress in this area. Besides first order logic, modal logic can be used to improve the theory. We plan to discuss also temporal logic for the relevant aspects of relativity.

Prerequisites. Before anything else, a good working knowledge of first order logic (FOL) is the most important prerequisite: model theory of FOL, many-sorted FOL. Some knowledge of linear algebra and fields would be desirable. Familiarity with first order modal logic and with the ideas of non-standard analysis would be useful, but not indispensable.

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