Coalgebra and Abstraction

Abstract

Frege’s Basic Law V and its successor, Boolos’s New V, are axioms postulating abstraction operators: mappings from the power set of the domain into the domain. Basic Law V proved inconsistent. New V, however, naturally interprets large parts of second-order $\mathsf{ZFC}$ via a construction discovered by Boolos in 1989. This paper situates these classic findings about abstraction operators within the general theory of F-algebras and coalgebras. In particular, we show how Boolos’s construction amounts to identifying an initial F-algebra in a certain category, we identify a natural coalgebraic dual to Boolos’s axiom which naturally interprets large parts of Aczel’s non-well-founded set theory via the construction of a certain terminal F-coalgebra, and we suggest a coalgebraic way forward for an abstraction-theoretic axiomatization of the real numbers.