Two Undecidable Problems of Analysis

Abstract

In this article two undecidable problems belonging to the domain of analysis will be constructed. The basic idea is sketched as follows: Let us imagine an area B of functions (rational functions, trigonometric and exponential functions) and certain operations (addition, multiplication, integration over finite or infinite domains, etc.) and consider the smallest quantity M of functions which contains B and is closed with regard to the selected operations. The question will then be examined whether there is in M a function f( x)for which the predicate P( n)≡ � f( x)cos nxdx > 0 is not recursive. It will be shown that by suitably choosing the area B and the operations, the answer comes out positively. We will deal in general with complex functions of real variables, although one could with somewhat more effort carry out all considerations in the real domain. In the first example, new functions will be generated by means of the following operations: addition, multiplication, integration over finite intervals and the solution of Fredholm integral equations of the second kind. Following this, it will be shown that certain logically characterised functions can be represented as limits of functions of the area M. In these constructions care will be taken that the number of integral equations to be solved remains as small as possible (namely two). In the second example, instead of the solution of Fredholm integral equations, we permit integration over infinite intervals, and then prove for this instance the same theorems as in the first example, in the context of which considerable use will be made of the result of M. Davis, H. Putnam and J. Robinson (cf. (1)) on the unsolvability of exponential diophantine equations.