What is so special with the powerset operation?

The powerset operator, P, is an operator which (1) sends sets to sets,(2) is defined by a positive formula and (3) raises the cardinality of its argument, i.e., |P(x)| > |x|. As a consequence of (3), P has a proper class as least fixed point (the universe itself). In this paper we address the questions: (a) How does P contribute to the generation of the class of all positive operators? (b) Are there other operators with the above properties, “independent” of P? Concerning (a) we show that every positive operator is a combination of the identity, powerset, and almost constant operators. This enables one to define what a P-independent operator is. Concerning (b) we show that every P-independent bounded positive operator is not P-like. 2000 Mathematics Subject Classification. Primary 03E05, secondary 03E20.