Measuring the Fuzziness of Sets

Abstract

Set theory begins to be useful when there is some natural criterion for defining belonging to a set. Sets of objects without properties are uninteresting. Elements are assigned to sets because they share properties or conform to a rule. A set of elements is said to be fuzzy when we allow some elements to belong to the set unequally or more strongly than others because they have more of the common properties. What we are doing here is to question the concept of equality of belonging of elements to sets. For example, the set of all red roses admits a wide range of redness even though some roses are more red than others. For some purposes, redness may be allowed to range from magenta red to light pink. For other purposes, the range of red may be narrow. If one wishes to be precise, one would have to measure redness in Angstrom units and admit in each set only those uniform roses (if such exist) which have that precise redness wavelength.

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