oSets: Observer-Dependent Sets

Sets play a foundational role in organizing, understanding, and interacting with the world in our daily lives. They also play a critical role in the functioning and behavior of social robots and artificial intelligence systems, which are designed to interact with humans and their environments in meaningful and socially intelligent ways. A multitude of non-classical set theories emerged during the last half-century aspiring to supplement Cantor’s set theory, allowing sets to be true to the reality of life by supporting, for example, human imprecision and uncertainty. The aim of this paper is to continue this effort of introducing oSets, which are sets depending on the perception of their observers. Our main objective is to align set theory with human cognition and perceptual diversity. In this context, an accessible set is a class of objects for which perception is passive, i.e., it is independent of perception; otherwise, it is called an oSet, which cannot be known exactly with respect to its observers, but it can only be approximated by a family of sets representing the diversity of its perception. Thus, the new introduced membership function is a three-place predicate denoted ∈i, where the expression “x∈iX” indicates that the “observer” i perceives the element x as belonging to the set X. The accessibility notion is related to perception and can be best summarized as follows: “to be accessible is to be perceived”, presenting a weaker stance than Berkeley’s idealism, which asserts that “to be is to be perceived”.