Interpretation and Truth in Set Theory

The present paper is concerned with the presumed concrete or interpreted character of some axiom systems, notably axiom systems for usual set theory. A presentation of a concrete axiom system (set theory, for example) is accompanied with a conceptual component which, presumably, delimitates the subject matter of the system. In this paper, concrete axiom systems are understood in terms of a double-layer schema, containing the conceptual component as well as the deductive component, corresponding to the first layer and to the second layer, respectively. The conceptual component is identified with a criterion given by directive principles. Two lists of directive principles for set theory are given, and the two double-layer pictures of set theory that emerged from these lists are analyzed. Particular attention is paid to set-theoretic truth and the fixation of truth-values in each double-layer picture. The semantic commitments of both proposals are also compared, and distinguished from the usual notion of ontological commitment, which does not apply. The approach presented here to the problem of concrete axiom systems can be applied to other mathematical theories with interesting results. The case of elementary arithmetic is mentioned in passing.