Difficulties of the set of natural numbers

TL;DR

This paper explores the foundational difficulties in defining the set of natural numbers, suggesting it may be a proper class rather than a set due to logical conflicts encountered during an iterative deduction process.

Contribution

It introduces a novel argument that challenges the set-theoretic conception of natural numbers using transfinite recursion and logical analysis.

Findings

Identifies difficulties in deducting all natural numbers via iterative processes.

Suggests the class of natural numbers is a proper class, not a set.

Highlights potential contradictions with the axiom of regularity.

Abstract

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.