Existence of a Total Order in Every Set

Problem statement: The axiom of choice, guarantees that all set could be well-ordered, in particular linearly ordered. But the proof in this case was not effective, that was to say, non constructive. It was natural to ask if there was ma thematics in which we could given a more constructive proof. Approach: We work in the Nelson’s IST which was an extension of ZFC (Zermelo-Fraenkel set theory with the axiom of choi e). In the theory of IST there were two primitive symbols st, ∈ and the axioms of ZFC together with three axiom sc hemes which we call the Transfer principle (T), the principle of Idealization (I) an d the principle of Standardization (S). Results: In the framework of IST we could construct, without the us e of the choice axiom, a total order on every set. Conclusion: The Internal Set Theory provides a positive answer to our question .


INTRODUCTION
The axiom of choice (Fraisse, 2000;Hrbacek and Jech, 1999) is an important and fundamental axiom in set theory sometimes called Zermelo's axiom of choice. It was formulated by Zermelo in 1904 and it differs from the other principles of set theory in that it is not effective, that is, a proof requiring the axiom of choice is nonconstructive.
This study is placed in the framework of IS T (Diener and Diener, 1995;Nelson, 1977, Nelson andRogers, 1988;Berg, 1992;Vath, 2007). Moreover, to prove the result announced above in the abstract, the principal tool is the use, instead of an infinite set X , of a finite subset F = {x 1 , x 2 ,…, x ω }⊂X containing all standard elements of X, transfer principle and standardization principle.

MATERIALS AND METHODS
In the proof of the result announced in the abstract, we use the following tools: • Theorem of Nelson (1977) that affirms: For all set X there is a finite set F⊂X such that for all standard x∈X we have X∈F. This theorem is one of the most interesting consequences of the Idealization principle (I) • Standardization principle (S) which one uses, generally, to create a new mathematical objects, that is to say, it is a principle of construction • Transfer principle (T) who has been used several times

RESULTS
Theorem: On every non empty set X we can construct a total order R. In particular, this total order can be chosen in such a way that X has a least element or at least and a greatest element.
Proof: Let X be any non empty set. Suppose, by transfer, that X is standard. If X is finite, then we can easily prove the theorem. Indeed, if X = {α 1 , α 2 ,…, α m } is a standard finite set, then every element of X is standard and m is standard (Diener and Diener, 1995;Nelson, 1977) and we can, for instance, define on X the following relation . Clearly, this relation is a total order. Moreover, the least element is α 1 and the greatest element is α m . Suppose that X is infinite. Let F = {x 1 , x 2 ,…, x n } be a finite subset of X containing all standard elements of X (Diener and Diener, 1995). Let be a subset of X×X and let G s be the standardization of G. It is known that G s defines a relation R from X to X as follows: We prove that X is totally ordered by R. Indeed, the relation is standard because G s is also. On the other hand, by construction, st x X Hence R is reflexive. Let x, y and t be three standard elements of X, such that xRy and yRt. Then, x, y and t are elements of F and This implies that in F the index of x is less than or equal to the index of y which is less than or equal to the index of t. Hence, the index of x is less than or equal to the index of t, which shows that (x, t) ∈G and therefore (x, t) ∈G s i.e. xRt. So we proved: x, y, t X xRy and yRt xRt ∀ ∈  ⇒    Then by transfer: x, y, t X xRy and yRt xRt ∀ ∈  ⇒    Which implies that R is transitive. Let x, y be standard elements of X such that xRy and yRx. Then, x, y is elements of F and belongs to G s i.e. (x, y), (y, x) belong to G.
This implies that in F the index of x is less than or equal to the index of y and inversely. Hence, the index of x equal to the index of y, which shows that x = y and consequently R is anti-symmetric. Thus R is a partial order.
Prove that R is a total order on X. Let (y, z) be a standard element of X 2 , then, y, z are standard elements of F. Hence, (y, z) ∈G or (z, y) ∈G because in F we have one and only one of the two following cases: the index of y is less than or equal to the index of z or inversely.
This signifies that ( ) x X x Rx ∀ ∈ which shows that x 1 is a least element for X. Analogously, so that R has a least element and a greatest element we choose x 1 and x N two standard elements. By transfer, we conclude for all X.

Remark:
Recall that by induction it is proven that "Every finite system of sets has a choice function" (Hrbacek and Jech, 1999). Therefore, when the system of sets is finite the existence of the choice function is provable and does not need to be deducted by an axiom. Let us show in the following how, in the previous proof, we choose x 1 standard then x 1 and x N standard.
x 1 standard. Consider in this case the finite system of sets {X} which is standard. Then there exists a choice function f 1 which is by transfer standard such that f 1 (X) ∈X. Put x 1 = f 1 (X). Then x 1 is by transfer standard. Let us take x , y , y ,..., y . Thus, F is a finite set such that the element having the smallest index (i.e. The index equals to 1) is standard.
x 1 and x N are standard. As in the case (a), for the system {X} there exists a choice function f 1 which is by transfer standard such that the element ( ) x , y , y ,..., y , x . Thus, F is a finite set such that the element having the smallest index (i.e. the index equals to 1) is standard and the element having the greatest index is also standard.

DISCUSSION
The absence of the axiom of choice, in our abovestated theorem, called the three axioms of the IST. This shows the fundamental role that plays this axiom in the set theory. Moreover, the previous techniques are always valid to get other results.

CONCLUSION
The Internal Set Theory has been adopted by quite a number of working mathematicians and this number becomes more and more important. This because it appears to be an efficient tool in several problems of mathematics, where we notice that most of proofs are characterized by the fact that they are constructive.
In our case the most decisive tool, next to the other principles of IST, is the replacement of an infinite set by one of its finite parts which contains all its standard elements. In addition of this, we also find that the standardization principle is of a particular importance because it gives birth to new objects.