Difficulties of the set of natural numbers

الأصل · EN

In this article some difficulties are deduced from the set of natural numbers. By using the method of transfinite recursion we define an iterative process which is designed to deduct all the non-greatest elements of the set of natural numbers. But unexpectedly we meet some difficulties in answering the question of whether the iterative process can deduct all the elements of the set of natural numbers. The demonstrated difficulties suggest that if we regard the class of natural numbers as a set we will be confronted with either a contradiction or a conflict with the axiom of regularity. As a result, we have the conclusion that the class of natural numbers is not a set but a proper class.