Abstract.
The powerset operator, 
, 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., |(x)|>|x|. As a consequence of (3), has a proper class as least fixed point (the universe itself). In this paper we address the questions: (a) How does contribute to the generation of the class of all positive operators? (b) Are there other operators with the above properties, “independent” of ? 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 -independent operator is. Concerning (b) we show that every -independent bounded positive operator is not -like.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aczel, P.: Non-Well-Founded Sets. CSLI Lecture Notes, Stanford, 1988
Feferman, S.: Why the programs for new axioms need to be questioned. Bull. Symb. Logic 6, 401-413 (2000)
Moschovakis, Y.: Elementary induction on abstract structures. North Holland P.C. 1974
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics Subject Classification (2000): Primary 03E05, secondary 03E20
Rights and permissions
About this article
Cite this article
Tzouvaras, A. What is so special with the powerset operation?. Arch. Math. Logic 43, 723–737 (2004). https://doi.org/10.1007/s00153-004-0229-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-004-0229-z