Proof. Myhill has proved that any productive set is productive under a 1-1 total function. A corresponding result is easily established for any contraproductive set having at least one total contraproductive function. Letting h(x, y) be the recursive function of [2, Theorem 2.4, p. 69], consider the sequence s0 = h(0, 0), Sx — h(l, s0), s2 = h(0, Sx), Si = h(i, s2), Si = h(2, S3), and so on. Th...

In this paper, following the ideas of (continuous) t-norm and interval numbers, a concept of (continuous) interval-valued t-norm is proposed. Based on the interval-valued fuzzy set and the continuous interval-valued t-norm, we propose a notion of interval-valued fuzzy metric space, which is a generalization of fuzzy metric space in the sense of George and Veeramani [Fuzzy Sets and Systems 64 (1...

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 ...