In practical life, researchers aim to appropriately frame societal problems and challenges to address and find effective solutions. One efficient method for managing complex real-world data is rough set theory. Utilizing rough approximation operators, it identifies both confirmed and possible data obtainable through subsets. Earlier studies have introduced several rough approximation models inspired by neighborhood systems, which aim to enhance accuracy and satisfy the axioms of traditional approximation spaces as initially proposed by Pawlak. In this work, we put forward novel paradigms of rough sets depending on the cardinality rough neighborhoods and Ideals. These models are a suitable approach to cope with a wide range of examples including issues related to cardinal numbers, which are frequently encountered in contexts such as social media engagement, visitor counts at exhibitions, and the evaluation of applicants based on the number of their qualities. We amply investigate the master features of these paradigms and elucidate the interrelations between them as well as their connection with previous ones. Then, we tackle these paradigms from a topological view as an alternative instrument for describing the boundary regions and calculating the accuracy of data. Moreover, we examine our models’ efficiency in dealing with dengue disease for some patients and conclude that the proposed rough-set paradigms ameliorate the properties of the previous approximation spaces. Ultimately, we demonstrate their pros in terms of expanding the confirmed knowledge obtained from subsets of data and keeping the main characteristics of original paradigms by Pawlak that were violated by forgoing models, as well as list the deficiencies of the present paradigms.