Existence of a Total Order in Every Set

Abstract

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 mathematics 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 choice). In the theory of IST there were two primitive symbols st, ∈ and the axioms of ZFC together with three axiom schemes which we call the Transfer principle (T), the principle of Idealization (I) and the principle of Standardization (S). Results: In the framework of IST we could construct, without the use of the choice axiom, a total order on every set. Conclusion: The Internal Set Theory provides a positive answer to our question.