An Alternative Set Model of Cognitive Jump

When we enumerate numbers up to some specific value, or, even if we do not specify the number, we know at the same time that there are much greater numbers which should be reachable by the same enumeration, but indeed we also congnize them without practical enumeration. Namely, if we deem enumeration to be a way of reaching a number without any "jump", there is a "jump'' in our way of cognition of such greater numbers. In this article, making use of a set theoretical framework by Vop\v{e}nka (1979) (alternative set theory) which describes such structure, we attempt to shed light on an analogous sturucture in human and social phenomenon. As an example, we examine a problem of common knowledge in electronic mail game presented by Rubinstein (1989). We show an event comes to common knowledge by a "cognitive jump".