Sentences Undecidable in Formalized Arithmetic : An exposition of the Theory of Kurt Gödel: Andrzej Mostowski

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This is an up-to-date exposition of Gödel's theory on the incompleteness of functional calculi which include ordinary arithmetic. In spite of the author's modest remarks, no strictly comparable work exists, at least to the reviewer's knowledge. There are indeed other books which contain chapters on Gödel's works, but none is entirely focussed on it like the present monograph. Apart from that, the author makes an original contribution to the subject by providing a unified treatment of the various methods of approach which have been put forward.

The contents of the book are as follows. The introduction contains a largely informal discussion of the notion of incompleteness and a sketch of the fundamental idea of Gödel's argument. Following certain auxiliary arithmetical theorems and recursive definitions, the object language $(S)$ is introduced and arithmetised. $(S)$ and its syntax are then investigated in detail. There follows a discussion of the semantics of $(S)$, based on Tarski's work. To achieve a unified treatment of the subject the author then introduces the notion of definability with respect to any given class of statements which is consistent and deductively closed. This includes Tarski's notion of definability and the notion of general recursivity as special cases. Gödel's main theorem is given in the last chapter with the extensions and corollaries of Rosser and Tarski. Gödel's ``second incompleteness theorem'' is given in the appendix. There follows a discussion of the problem, first reviewed by Gödel and considered in more detail by Tarski and others, how to extend the incomplete calculus of $(S)$ in order to derive the theorem which was previously unprovable. [Reviewer's comment. The addition of supplementary rules ad hoc---and indefinitely---does not solve the problem whether there is a criterion which would replace the ``intuitive'' opinion of all mathematicians, who are not intuitionists, that a given type of mathematical argument is acceptable. The fact is that such almost universal agreement exists for all mathematical theories developed so far.] The book concludes with a section based on Gödel's ideas on the length of proofs.

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