The following article is an axiomatic approach to defining and describing interval-division numeral systems.
Interval-Division Numeral Systems (PDF)
In addition, I have included a supplemental article that proves the details that I did not prove explicitly in the original article.
Interval-Division Numeral Systems (Supplement) (PDF)
Interval-division numeral systems are the most general class of numeral systems that we will be considering in this topic. They are so named because they can be thought of as the class of numeral systems described by repeatedly dividing the unit interval into smaller subintervals, and using membership in those subintervals as an address scheme uniquely describing points on the unit interval.
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