Abstract
A perfect theory for new fuzzy sets, called Zadeh sets is outlined. Here are four observations, one theorem, and the proposed theory. (1) Fuzzy sets are not fuzzy - An ancient critique. (2) The theory of fuzzy sets does not support fuzzy control - Example 1. (3) The concept of fuzzy sets induced by mappings is not natural-Section II-A. (4) Zadeh sets should be context free - Section III-A. (5) Multi-valued modeling Theorem A number can be modeled uniquely by a family of smooth(C^∞-differentiable) membership functions. By extending this idea, we propose: A nice Zadeh set is the unique family of nice membership functions on a nice universe U that characterizes a nice real world fuzzy aggregate (class, family, collection or set) where nice means smooth(C^∞-differentiable), continuous or set theoretical (a set theoretical membership function is the usual membership function defined by Zadeh.) Initial analysis indicates that such three types of nice Zadeh sets seem to have captured the correct concept of fuzziness. In addition, mathematically speaking, the theory forms naturally, behaves smoothly and three categories of nice Zadeh sets are concrete categories. So we conclude Z adeh set theory is perfect. Perhaps, we should point out the 'classical' categories of fuzzy subsets are not concrete.
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