The validity of the properties of real numbers set to hyperreal numbers Set

This research discusses the validity of the properties of real numbers set to hyperreal numbers set, i.e. algebraic, ordered, and completeness properties, by using a finitely additive measure. This finitely additive measure is a map from the power set of natural numbers set onto set {0,1}. The subset of natural numbers set has measure zero if it’s finite and one if it’s infinite. The set of hyperreal numbers is constructed from equivalence classes of the set of all sequences of real numbers by using a relation involving the finitely additive measure, that is, two sequences of real numbers are said to be related if and only if those two sequences are the same almost everywhere. In the hyperreal numbers set, there exist infinitesimal numbers besides 0. Infinitesimal number is a number which is less than any positive real number and greater than any negative real number. So, in hyperreal number set, there are some smallest positive numbers. The results show that the hyperreal numbers set is an ordered and complete field.


Introduction
The set ℝ of real numbers can be constructed by using the concept of equivalence classes of the set of all rational Cauchy-sequence (see [3]). Abraham Robinson [2] had expanded the real numbers set. This expansion is called hyperreal numbers set * ℝ. This numbers set is constructed by using the concept of equivalence classes of real numbers sequence. For example, both   1/ n and   2 1/ n are the sequences that converge to zero, but have different convergence rates. In * ℝ, convergence rate is also considered. Then, in * ℝ, both sequences are in different classes.
Another interesting property of hyperreal numbers set * ℝ is there exist more than one "small" or infinitesimal number. Infinitesimal number is a number which is less than any positive real numbers and greater than any negative real numbers. Differently from hyperreal numbers set * ℝ, real numbers set ℝ has one infinitesimal number, that is zero. Moreover, in * ℝ, there is a "quantification" concept for "large" or infinite numbers. In * ℝ, infinite property can be manifested as a number. Therefore, this paper investigates whether or not algebraic, ordered, and completeness properties, which are valid in real numbers set ℝ (see [1]), are also valid in hyperreal numbers set * ℝ.

Methods
The study of the validity of the properties of real numbers set ℝ to hyperreal numbers set * ℝ is performanced by the following steps.
1. Constructing hyperreal numbers set using the concept of equivalence classes and measure : (ℕ) → {0,1} with (ℕ) is the power set of natural numbers set ℕ. 2. Studying the properties of real numbers set which are valid in hyperreal numbers set, such as algebraic, ordered, and completeness properties by using the measure m.

Results and Discussion
Hyperreal numbers set * ℝ is constructed by using the concept of equivalence classes in the set of real numbers sequences (see [2]).

Construction of hyperreal numbers set * ℝ
Hyperreal numbers set is constructed of using a measure : (ℕ) → {0,1} with (ℕ) is the power set of natural numbers set ℕ which is defined by the following definition.
The measure m is finitely additive measure which means 

for all disjoint sets
A and B. The measure m divides the subset of ℕ into two parts, those are, a "large" or infinite set with a measure one and a "small" or finite set with a measure zero.
Any subset ⊂ ℕ satisfies one of the conditions   1  mA . For any , ⊆ ℕ with . Next discussion is the construction of * ℝ by using the concept of equivalence classes on the set of all real numbers sequences.

Definition 3.2
Let  is the set of all real numbers sequences and ∼ is an equivalence relation on  which is defined by the following. For all     In other words,   n a is the same as   n b almost everywhere. This following is the definition of hyperreal numbers set by using equivalence relation ∼. (2) Next, hyperreal numbers set * ℝ is a set of all equivalence classes in set  and is denoted by To study the properties of hyperreal numbers set which is analogous to real numbers set, the addition, multiplication, and order operation on * ℝ are defined.

Definition 3.4 For each
,  nn ab * ℝ, the addition operation "  ", multiplication operation "  ", and order Next, each ∈ ℝ can be written as *  n aa The set * ℝ is divided into three parts, those are infinitesimal numbers, finite numbers, and infinite numbers.

Example 3.1
The numbers * 0 , *   has a unique inverse element and the multiplication of two numbers in * ℝ that produces * 0 is fulfilled when one or both of these numbers is * 0 .  e) Using the same way as in proof of d), we can prove that if * a ** 0  b , then ** 0  a and ** 0  b , or ** 0  a and ** 0  b . ■ In real numbers set, natural numbers are positive real numbers [1]. Also, natural numbers in hyperreal numbers set are positive hyperreal numbers. Moreover, in real numbers set, there is no smallest positive real numbers [1]. In hyperreal numbers set, there exist infinitesimal numbers, for example 1/ , n since     : 1/ 1     m n a n a for each ∈ ℝ + . Therefore, in hyperreal numbers set, there are smallest positive hyperreal numbers.

Completeness
Properties. This following is a discussion of completeness properties of * ℝ, that is the completeness properties which is related to the supremum and infimum concepts in * ℝ.

Definition 3.8
Let * A is a non-empty subset of * ℝ.