Abstract
This chapter discusses the interval components of non-archimedean number systems. Real ordinal numbers with positive diameter arise on the basis of linear transfinite rational balls of numbers. The chapter also discusses those problems in this area that focus upon interval-mathematical ideas. The domain of ordinal numbers is well-ordered by a non-greater relation and irreflexively well-ordered by a smaller relation. The usual addition and multiplication of ordinal numbers are neither commutative nor absorption-free. However, they are not suitable for the construction of a transfinite number system consisting of integral, rational, and real ordinal numbers. The chapter also describes the rational basis of a real extension of transfinite number systems. The transfinite rational domains differ essentially from the classical rational domain.
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