On Recursive Unsolvability of Hilbert's Tenth Problem

Abstract

This chapter discusses recursive unsolvability of Hilbert's tenth problem. The tenth problem is the only one of the 23 problems that clearly has an algorithmical nature. The problem proved to be rather difficult and only last year it was shown to be unsolvable. There is no algorithm for determining whether an arbitrary diophantine equation has a solution. The chapter discusses history of how the unsolvability of Hilbert's tenth problem was proved. Every recursively enumerable (r.e.) predicate is diophantine. The classification of arithmetical formulas with bounded universal quantifiers is defined.