Undecidable Theories

Undecidable Theories

Undecidable Theories

Undecidable Theories

Excerpt

This monograph consists of three papers: A general method in proofs of undecidability, Undecidability and essential undecidability in arithmetic, Undecidability of the elementary theory of groups.While the first and the third papers have been written by the undersigned alone, the second paper is a joint work of A. Mostowski, R. M. Robinson, and the undersigned.

The three papers are referred to throughout the monograph by Roman numerals I, II, III. The introduction to paper I is thought of as an introduction to the whole work and gives an idea of the scope of the problems discussed and the results obtained. The notations and symbolic conventions introduced at any place of the monograph are applied in the whole subsequent discussion.

The work contains results obtained over a long period of time,1938-1952. The first draft of paper II was prepared by Mostowskiin 1949 and contained exclusively results found by him and Tarskiin the pre-war period. However, its publication was postponed and the text was considerably modified, so as to embody the results and simplifications subsequently obtained by Robinson and partly by Tarski; the final draft was written jointly by these two authors in 1951-52. The other two papers were prepared for publication during the same period.

It was originally planned to publish the work as a series of connected papers in some regular mathematical periodical. However, the Editors of the series Studies in Logic found the material integrated enough to appear as a separate volume in this series; their offer was accepted with appreciation.

A large part of the technical work on the monograph was done during the period when Tarski and, for a shorter time, also Robinson were engaged in a research project in the foundations of mathematics sponsored by the Office of Ordnance Research, U.S. Army.

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