Finiteness notions in fuzzy sets

Finite sets are one of the most fundamental mathematical structures. In the absence of the axiom of choice there are many different inequivalent definitions of finite even in classical logic. When we allow incomplete existence as in fuzzy sets the situation gets even more complicated. This paper gives nine distinct definitions of finite in a fuzzy context together with examples showing how the properties of the underlying lattice of truth values impact the meanings of finite. © 2001 Elsevier Science B.V. All rights reserved. One of our most fundamental mathematical notions is that of finite set. When I told my colleague Narendra Jaggi, a physicist, the title of the talk this paper is based on his reaction was to say "Trust a mathematician to make the obvious difficult." In some sense that is the problem with finiteness: we are so used to working with finite collections of things in everyday life that the problem seems to be with what infinite means rather than with what finite means. But everyday life gives us a strong intuition about what finite means, not a rigorous definition we can use to provide a foundation for combinatorial mathematics and arithmetic. Difficulties with finiteness definitions in set theory have been known since the early twentieth century. Tarski's paper [14] gives the classical treatment of several of the variants treated here in the fuzzy case. Rubin's paper [ 10] gives an easily accessible modern exposition of the equivalence of many notions of finite, updating Tarski's paper. Jech's bookon the axiom of choice [6] shows how in the absence of the axiom of choice different definitions are inequivalent. Combinatorics is the branch of mathematics which deals with finite sets. Its main aim is to talk about how finite sets can be structured to make it possible to count their elements. There are three main principles of counting which form the basis for a starting place on what finiteness should mean: 1. If A n B = 0 then A U BA + IB I : disjoint cases lead to addition. 2. IA xB ALB: successive choices multiply. 3. If all equivalence classes for an equivalence relation N on A have the same number of elements m then IA/ A~/m: a systematic over count can be corrected by division. This tells us to expect that the class of finite sets should be closed under the operations of 1. Disjoint union (coproduct in the categorical setting). 2. Cartesian product (product in the categorical setting). Fuzzy Sets and Systems 161 (2010) 1162-1174 Elsevier www.elsevier.com/locate/fss When Does a Category Built on a Lattice with a Monoidal Structure have a Monoidal Structure?