Nicola Bonatti

Hilbert’s (arithmetical) Axiom of Completeness asserts that the structure of the real numbers $$\mathbb {R}$$ R is maximal in the sense of not having a proper extension to an Archimedean ordered field. The more recent works of Ehrlich (2001), McGee (1997) and Aczel (1988) show that certain maximality conditions modeled upon Hilbert’s axiom provide unique characterizations of, respectively, the s-hierarchical ordered field of surreal numbers No, the well-founded hierarchy of pure sets $$\mathbb {V}_{\!k}$$ V k, and the non-well-founded hierarchy of Finsler-extensional sets $$\mathbb {V}_{\scriptscriptstyle \!F\!A\!F\!A}$$ V F A F A. The paper provides a comprehensive historical and theoretical reconstruction of this often overlooked chapter in the history of the axiomatic method. The historical reconstruction suggests that the maximality condition of non-extensibility arises as a natural axiom in the unique characterization of mathematical structures, as illustrated by the theories of (s-hierarchical non-)Archimedean continua and (non-)well-founded sets. The theoretical reconstruction argues that the maximality condition of non-extensibility is the formal explication of the heuristic principle of Plenitude, as usually adopted to describe the intuitive ‘‘fullness” of punctiform continua and the cumulative hierarchy of sets.

Extremal axioms impose a condition of either minimality or maximality on the admissible models of an axiomatic theory. In this paper, I propose an alternative formulation of arithmetic and real analysis based on extremal axioms. Once properly formulated, the second-order extremal axiom restricts the quantifiers of the theory to the minimal or maximal domain of discourse. It is proved that extremal axioms are logically equivalent to standard assumptions of, respectively, second-order Induction and Archimedean Completeness. Finally, I distinguish between internalist and externalist accounts of mathematical structures as characterized by extremal axioms and their corresponding axiomatic theories.