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Introduction to mathematical logic. (English) Zbl 0192.01901

The University Series in Undergraduate Mathematics. Princeton, N.J.-Toronto-New York-London: D. Van Nostrand Company, Inc. x, 300 p. (1966).
The book is an expansion of lecture notes for a one-semester course in mathematical logic given by the author. The principal aim of mathematical logic is defined to be a precise and adequate definition of the notion of mathematical proof. The paradoxes are treated in the introduction in order to motivate the modern logical study.
Chapter I – Propositional calculus (among others with a section devoted to many-valued logics). Chapter II – Quantification theory. Up to completeness theorem, Skolem normal forms, categoricity in power, the 2nd \(\varepsilon\)-theorem. Chapter III – Formal number theory. Devoted to the questions of undecidability (Gödel-Rosser Theorem, non-definability of truth, undecidability of the predicate calculus).
Chapter IV – Set theory. Investigated is the Neumann-Bernays-Gödel system NBG (without axioms of regularity and choice). The reviewer agrees with the author that NBG is “probably the simplest and most convenient basis for the practicing mathematician”. Main topics: ordinal and cardinal numbers, equivalents of the axiom of choice, consistency of the axiom of restriction (regularity). The summary of the status of the consistency and independence of the axiom of choice and continuum hypothesis is, unfortunately, not up-to-date because the results of Cohen are not mentioned and the independence of the axiom from NBG + the axiom of restriction is claimed to be an open problem.
Chapter V – Effective computabilty. (Equivalence of Markov algorithms, Turing machines, Herbrand-Gödel computability and recursive functions. Undecidable problems.)
Appendix – The Schütte’s proof of the consistency of arithmetics.
Nowadays there are several introductory books on logics. It can be said that this book is a very good one, readable and compact. It is a very good introduction (with lots of examples and exercises) not containing too advanced topics but pointing out the way to them.

MSC:

03-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations