Abstract

Some partitions of Natural Number set are built through recursive processesgenerating in this manner countable examples of countable and disjoint sets whose unionis a set also countable. This process is constructive, so the Axiom of choice is not used.We provide a PC program that generates one of these special partitions and shows howto generate infinite of them. This line of reasoning can have multiple applications in Settheory and Model theory. We proved that the number of ways to make these partitionsof natural numbers is not countable, there are more of these partitions (named doublycountable) than natural numbers. For each natural number greater than 1, we show aneffective procedure that generates these partitions