Andréka, H. Madarász, J.X. and Németi, I.

On the logical structure of relativity theories

Abstract: This book has several aims. One of these is to make the theories of relativity more accessible, easily understandable for the reader having some familiarity with modern logic. (No knowledge of physics is presupposed.) One of the advantages of the logic based approach is that the reader does not have to believe anything we say: for any of the claims we make (e.g. moving clocks slow down), the reader can trace the logical proof we give for the claim to a few very simple and convincing axioms. Some relativity books are written in the ``popular style'' so they provide a wonderful reading but they do not prove the claims they make, therefore the reader has to believe what he is told without understanding why exactly this is being told. This renders many of these nice books like fairy tales. Other relativity books are technically precise but then they are extremely hard to read even for the average mathematician. The present book aims at having both of the good things: being easy to read but not being like a fairy tale. The reader need not believe anything (logically convincing proofs are provided) but we try to avoid using mathematical machinery which can be replaced by ``straight'' logical reasoning. There are further aims, these are listed in Section 1.1 of Part I.

This book consists of seven parts. Part I (approx. 150 pages) is of an introductory character and was designed to be easy to read (with intuitive pictures, drawings, motivating examples etc). Some familiarity with first-order logic is presupposed. The rest of the parts can be read independently of each other, but each part presupposes Part I.

E.g. the reader may read Part I and then may go directly to reading Part V (Chapter 6). Definitions used in Chapter 6 but not introduced in Part I are collected in Part VII which contains useful lists like list of axioms, index, bibliography.

Part I consists of Chapters 1,2. The rest of the chapters count as individual parts. E.g. Chapter 3 coincides with Part II.

Part VII consists of various lists designed to help the reader reading Parts II-V independently of each other.

Part VI contains a detailed explanation of why we insist on using first-order logic and its variants instead of, say, second-order logic. It is called appendix. Parts I, III, V (Chapters 1,2,4,6) are in polished, hopefully final form. (We plan to do some more polishing on Part I but not on Part V.) Parts II, IV, VI (Chapters 3,5, and the appendix) will be polished further, but are already readable.

Part I is a general introduction and a general overview. Among other things, Part I shows how one can build up the theory of special relativity (and its variants) as a purely logical theory in the framework of first-order logic. (Later parts contain generalizations toward general relativity, but still staying in first-order logic.) However, as the introduction to Part I shows, the present work is much more ambitious than building up certain existing theories of physics in logic.

Part II (Chapter 3) introduces a new, more flexible (than the one in Part I) theory called Newbasax. It discusses the existence and properties of the models of the theories introduced in Parts I, II.

Part III (Chapter 4) elaborates the ``lego'' character of our approach, it shows how one can modify the (relativity) theories by replacing axioms, and it studies consequences of such modifications (e.g. do the so called paradigmatic effects of relativity remain provable). It also provides a logically precise comparison of the various distinguished versions of relativity (like Einstein's one, Reichenbach's one, etc).

Part IV (Chapter 5) shows how to derive (in logic) relativity theory without mentioning photons or electrodynamics at all. This part uses ideas from Gyula David (ELTE Univ., Dept. Physics).

Part V (Chapter 6) elaborates the ``geometrization'' of relativity theory leading up to e.g. geodesics and other notions indispensable for general relativity. This part contains precise (in the math. logic sense) investigations of methodological issues like the theoretical/observational distinction about the concepts one uses, what formulas count as potential laws of nature, methodological ideas ranging from Occam's razor, Leibniz, Kant, the logical positivists, Reichenbach, Mach, Einstein, Gödel. Special attention is given to Gödel's ideas about relativity and its philosophical consequences (since Gödel was one of the greatest logicians of all times and his ideas about relativity and its philosophical significance/consequences are not as well known as those of e.g. Mach or Einstein). Also, consequences of Gödel's incompleteness theorems for relativity theories show up in this part.

Part VI (Appendix) contains detailed explanation of why we have to stick with first-order logic or its variants like many-sorted modal first-order logic as opposed to higher-order logic, in a work whose goals are like those of the present one.