Alternative set theory (AST), or the theory of semisets, the spiritual father of whom is the well-known mathematician Petr Vopěnka, has never received wide recognition and at the present time is subject to neglect. In spite of this, however, it is a conception which is notable for its absolute straightforwardness and for the fundamentally different way in which it mathematises the concept of indeterminacy: the mathematisation of the infinite can be interpreted as an absolutely fundamental mathematisation of the indeterminate. A key role is played by an undetermined grouping – so called semisets. In this article, examples of semisets are discussed in detail and it is shown that semisets should be understood consistently as potentially infinite sets in the spirit of Aristotelian potential infinity. In other words, as sets contain in them a permanent possibility of creating or discovering more and more objects belonging to the given grouping.