The Re-evaluation of the Euclidean Number System


Dedekind Cut

    Representing the length is the important usage of the number. In this case, we must determine the unit of length. Subsequently, we can measure any length as the ratio to the unit length. If we ignore the error, this method is sufficient for the practical usage. Rather, it is necessary to ignore the error in the practical calculation. However, ancient Greeks discovered the length, which does not have the ratio to the unit. Though this discovery was the big shock for them, they found a solution. That is, Euclid distinguishes the number from the magnitude in the Elements. A Euclidean number has to consist of units. So, the Euclidean number and the practical calculation are the good match. Let us re-evaluate the Euclidean number system and compare it with the modern number system.     Dedekind defined a real number as a Dedekind cut. An arbitrary real number ƒ¿ divides rational numbers into two classes: (A1,A2). Any element of the class A1 is greater than that of the class A2. (A1,A2) represents the Dedekind cut, which corresponds to ƒ¿. Consider two Dedekind cuts (A1,A2) and (B1,B2). They are produced by two real numbers ƒ¿ and ƒÀ, respectively. Let us compare these two Dedekind cuts. The results are divided into three cases.
  1. If ƒ¿ is greater than ƒÀ, A1 contains at least two different rational numbers, which are not contained in B1.
  2. If ƒÀ is greater than ƒ¿, B1 contains at least two different rational numbers, which are not contained in A1.
  3. If ƒ¿ equals ƒÀ, the results are divided into two cases. The first case, A1 is identical to B1. The second case, the difference between A1 and B1 is only one rational number, which equals ƒ¿ and ƒÀ. That is, these two Dedekind cuts are essentially identical.

Counting Dedekind Cuts

    Next, we shall consider whether the number of irrational numbers is larger than that of rational numbers. Let us regard an irrational number as a pigeon and an interval of two rational numbers as a pigeonhole. According to the pigeonhole principle, if the number of irrational numbers were larger than that of rational numbers, there would be at least two irrational numbers, between which there is no rational number. However, there is no such a pair of irrational numbers, because there are at least two rational numbers between two different Dedekind cuts. That is, the number of irrational numbers is not larger than that of rational numbers. This is the first conclusion.
    Next, we shall consider another proof. Consider a real number ƒ¿ and the larger real number ƒÀ, and the difference of them equals ƒÃ:
According to the axiom of Archimedes, no matter how small ƒÃ is, there exists a natural number  such that:
Then, there exists an integer m such that:
That is, there must be a rational number between two different real numbers. In other words, any real number is sandwiched between two rational numbers. Because real numbers contain rational numbers and irrational numbers, the number of irrational numbers is not larger than that of rational numbers. This result coincides with the first conclusion.

Contradictions

    Next, the same logic as above can be applied to (m/n)ƒÎ. Then, there exists an integer m and a natural number n such that:
That is, the number of the irrational numbers, which equal ƒÎ times a rational number, is not smaller than that of all other real numbers. This is the second conclusion. However, if the first conclusion is true, all irrational numbers must be rational multiples of ƒÎ. It is impossible. Therefore, the second conclusion contradicts the first conclusion.
    Furthermore, consider an arbitrary real number ƒÁ. The same logic as above can be applied to (m/n)ƒÁ. Then, there exists an integer m and a natural number n such that:
That is, the number of the real numbers, which equal ƒÁ times a rational number, is not smaller than that of all other real numbers. If we substitute an irrational number for ƒÁ, we can obtain one more conclusion. Hence, we can obtain an arbitrary number of conclusions. These conclusions contradict the first conclusion.

The Euclidean Number System

    These contradictions mean that there is an error in the preconditions. At such case, it is best to return to the classic. We shall reread Euclid's Elements . Perhaps, Euclid might notice these contradictions, and then he described the solution. Firstly, a magnitude is defined as a length of a line segment in Book ‡X. Secondary, Euclid says "A number is a multitude composed of units" in the definition 2 of Book ‡Z. The difference between Euclid and Dedekind is evident. Dedekind regarded a magnitude as a number. In contrast, Euclid distinguishes the number from the magnitude. If we accept Euclid's standpoint, contradictions are eliminated.
    For an example, if we define ƒÎ as a unit, the magnitude nƒÎ corresponds to the natural number N:
In this case, one-to-one correspondence can be applied to numbers but cannot be applied to other magnitudes. Next, we shall consider the magnitude Q, which has the ratio to ƒÎ. Q is represented by (m/n)ƒÎ:
Next, consider the magnitude R, which does not have a ratio to ƒÎ. In this case, Euclid described the concept equivalent to the Dedekint cut in the definition 5 of Book ‡X.  Hence, we can approximate R by (m/n)ƒÎ as precise as we want:
That is, we can handle any real number using Euclidean number system. Even if we define an arbitrary magnitude ƒÁ as the unit, we can approximate R to the required accuracy:
In the above equation, if we substitute 1 for ƒÁ, Dedekind cut is obtained. Then, we can define an irrational magnitude as the cut of rational magnitude. This is the special case. 

The Practical Usage of the Euclidean Number  

    Why does the number distinguish from the magnitude? If we want to count objects, we have to regard it as indivisible. Furthermore, if we have to count water, we pour water into bottles, and then we count bottles. That is, we can only count discrete and indivisible objects. At least, we have to regard the object as temporarily indivisible. Hence, we regard the magnitude, which consists of units, as the number.
    When we apply the Euclidean Geometry to the real world, we have to draw a line with a pencil or a printer. In both cases, there is the minimum dot. Even if we use any kind of tool, the minimum dot exists. So,  we can only constitute a line from dots. That is, only the magnitude, which equals the Euclidean number, can be realized. So, the decision of the unit is important. When we actually use the Euclidean number,  the unit is determined on a case-by-case basis. As a result, we can flexibly adapt the unit to the practical usage. Therefore, the Euclidean number system is sufficiently useful and does not cause contradictions.

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