Fraenkel’s axiom of restriction: Axiom choice, intended models and categoricity

A recent debate has focused on different methodological principles underlying the practice of axiom choice in mathematics (cf. Feferman et al., 2000; Maddy, 1997; Easwaran, 2008). The general aim of these contributions can be described as twofold: first to clarify the spectrum of informal justification strategies retraceable in the history of mathematical axiomatics. Second, to evaluate and to philosophically reflect on the actual reasoning involved in the introduction of new axioms in mathematical practice such as large cardinal axioms in set theory. The most extensive treatment of these matters for the case of set theory can be found in (Maddy, 1997). Her philosophical discussion of axiom choice covers both of the mentioned approaches, i.e., it is both descriptive in reconstructing the justification types in early axiomatic set theory as well as normative in devising “methodological maxims” for the evaluation of present set theoretic axiom candidates. More specifically, Section 1.3 of her book provides a historical survey of the arguments given for ZFC by Zermelo, von Neumann, and Fraenkel (among others) with the intention “to explicate and analyze its distinctive modes of justification” given there (Maddy, 1997, p. 72). In the final two sections of her book (Sections 3.5 and 3.6), Maddy in turn devises “a naturalistic program” for discussing more recent axiom candidates (starting from Gödel’s axiom of constructibility to large cardinal or determinacy axioms) intended to model the “justificatory structure of contemporary set theory” (Maddy, 1997, p. 194). In this paper I attempt to take up Maddy’s historical discussion by drawing attention to a historical episode from early axiomatic set theory centered on Abraham Fraenkel’s axiom of restriction (“Beschränktheitsaxiom”) (in the following AR). The axiom candidate was first introduced by Fraenkel in the early 1920s and can be considered as a minimal axiom devised to