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 Nelsons 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.
Highlights
IntroductionThe 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
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
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 = {x1, x2,..., xω}⊂X containing all standard elements of X, transfer principle and standardization principle
Summary
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. 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 = {x1, x2,..., xω}⊂X containing all standard elements of X, transfer principle and standardization principle
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