Hilbert’s problems and logic. (Hilbertovi problemi i logika).

*(Serbo-Croat)*Zbl 0636.03001
Matematička Biblioteka, 48. Beograd: Zavod za Udžbenike i Nastavna Sredstva. 168 p. (1986).

At the Seminar of Mathematical Logic of the Mathematical Institute of Belgrade a sequence of lectures were dedicated to David Hilbert (1862- 1943), one of the greatest mathematicians of this century. The above mentioned lectures were presented on the occasion of 120 years of Hilbert’s birth and particular attention has been paid to four of Hilbert’s problems related to mathematical logic. This book is an extended version of those lectures.

In the introductory chapter, written by Mijajlović and Došen, some interesting historical data from Hilbert’s epoch are given, Hilbert’s short biography and the list of the 23 famous problems, presented by Hilbert at the Paris Congress of 1900.

The first chapter, written by Marković, deals with Hilbert’s first problem - Cantor’s continuum problem. After a short review of the importance and the place of this problem in the context of set theory, Gödel’s and Cohen’s independence theorems are proved.

Došen is the author of the second chapter. He treats Hilberts second problem - consistency of arithmetic, and gives Gödel’s incompleteness theorems and Gentzen’s proof of the consistency of arithmetic.

Hilbert’s tenth problem - solvability of Diophantine equations - is dealt with in the third chapter, written by Mijajlović. Matijasevič’s solution of this problem is presented and the theory of effective computability is outlined.

In the last chapter, Mijajlović considers Hilbert’s seventeenth problem: Let f be a positive definite rational function over the field of rational or real numbers. Is f necessarily a sum of squares of rational functions ? Robinson’s solution of the mentioned problem is offered.

It is a real pleasure to read such a book since the material contained is presented very clearly and correctly.

In the introductory chapter, written by Mijajlović and Došen, some interesting historical data from Hilbert’s epoch are given, Hilbert’s short biography and the list of the 23 famous problems, presented by Hilbert at the Paris Congress of 1900.

The first chapter, written by Marković, deals with Hilbert’s first problem - Cantor’s continuum problem. After a short review of the importance and the place of this problem in the context of set theory, Gödel’s and Cohen’s independence theorems are proved.

Došen is the author of the second chapter. He treats Hilberts second problem - consistency of arithmetic, and gives Gödel’s incompleteness theorems and Gentzen’s proof of the consistency of arithmetic.

Hilbert’s tenth problem - solvability of Diophantine equations - is dealt with in the third chapter, written by Mijajlović. Matijasevič’s solution of this problem is presented and the theory of effective computability is outlined.

In the last chapter, Mijajlović considers Hilbert’s seventeenth problem: Let f be a positive definite rational function over the field of rational or real numbers. Is f necessarily a sum of squares of rational functions ? Robinson’s solution of the mentioned problem is offered.

It is a real pleasure to read such a book since the material contained is presented very clearly and correctly.

Reviewer: B.R.Boričić

##### MSC:

03-02 | Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations |

03-03 | History of mathematical logic and foundations |

01A60 | History of mathematics in the 20th century |