Theorems Equivalent to Completeness Axiom

All principles and methods in mathematical analysis are based on the theory of real numbers and are taken as a starting point. Therefore, it is important to have a correct knowledge for the set of real numbers in order to have a profound knowledge of the principles and methods of analysis. There is a common point between two sets of numbers, called "ordered fields," in which both operations are defined and the size relation between any two numbers holds in the algebraic sense. However, there is an essential difference between the two sets. It cannot be described simply by algebraic operations, such as a set of integers from a set of natural numbers, a set of rational numbers from a set of integers. The essential feature of the set of real numbers compared to the set of rational numbers can only be explained correctly by the limit concept. In fact, for beginners of advanced mathematics, rational or real numbers have been frequently learned in their previous stages of education, but the essential characteristics of the two sets of numbers are not well understood and they are difficult to understand. We are trying to explain this more convincingly.