IT IS GENERALLY IMPOSSIBLE TO DETERMINE WHETHER A RATIONAL POINT LIES IN THE INTERIOR OR ON THE BOUNDARY OF A CLOSED SET ALGORITHMICALLY

Abstract

It seems true that we can almost always determine the position of a specific point when a set is given. Namely, we could assert whether this point lies at this set's interior, boundary, or exterior. However, this is not always the case in constructive mathematics. In this research, we will show that it is generally impossible to algorithmically determine whether a rational point lies in the interior of a closed productive set or on the boundary of it. We conduct our proof by making contradictions. Firstly, We used an unextendible algorithm to construct a rational point and a closed set on the natural line. Secondly, we reformulate the assumption "we could decide whether the point lies in the interior or on the boundary of a closed set” to “we could determine the program will eventually print 1". Thirdly, we constructed an extension of the program to all the positive integers, which is a contradiction to our assumption. Hence, we concluded that it is impossible to figure out the position of the rational point algorithmically.