Construction of the Transreal Numbers and Algebraic Transfields

The transreal numbers, introduced by James Anderson, are an extension of the real numbers. The four arithmetical operations of addition, subtraction, multiplication and division are closed on the set of transreal numbers. Transreal arithmetic has engendered controversy because it allows division by zero and is proposed as a replacement for real arithmetic. Anderson introduced the transreals intuitively and axiomatically. In the history of mathematics, constructive proofs have ended controversies. We construct the transreal numbers and transreal arithmetic from the very well accepted real numbers and real arithmetic. This construction proves consistency. We then extend the very well accepted algebraic structure of a field to a transfield. We show that, just as the rationals are the smallest, ordered field and reals are the unique, ordered, complete field, so, under suitable conditions, transrationals are the smallest, ordered transfield and transreals are the smallest, ordered, complete transfield. Thus we both prove consistency and demonstrate the wider applicability of the transreals. We hope this does enough to end controversy about the correctness of the transreals, leaving an assessment of their usefulness to future experience.

詹姆斯·安德森（James Anderson）提出的跨界数字是实数的扩展。加法，减法，乘法和除法的四个算术操作在经过超大数字的集合上关闭。 Transreal算术引起了争议，因为它允许零划分，并被提议作为实际算术的替代。安德森（Anderson）以直觉和公理的形式引入了特写。在数学历史上，建设性证据结束了争议。我们从非常公认的实际数字和实际算术中构造了经过的经过实现数量和经过超实算术。这种构造证明是一致性的。然后，我们将磁场的非常公认的代数结构扩展到了跨场。我们表明，正如理性是最小的，有序的字段，而实数是独特的，有序的，完整的字段，因此，在适当的条件下，跨元素是最小的，有序的变形底场，而经过的特性是最小的，有序的，完整的，完整的transfield。因此，我们都证明了一致性，并证明了经过特性的更广泛的适用性。我们希望这足以结束有关特雷尔斯的正确性的争议，并评估其对未来经验的有用性。