The iterative conception of set A (bi-)modal axiomatisation

The use of tensed language and the metaphor of set ‘formation’ found in informal descriptions of the iterative conception of set are seldom taken at all seriously. Both are eliminated in the nonmodal stage theories that formalise this account. To avoid the paradoxes, such accounts deny the Maximality thesis, the compelling thesis that any sets can form a set. This paper seeks to save the Maximality thesis by taking the tense more seriously than has been customary (although not literally). A modal stage theory, MST, is developed in a bimodal language, governed by a tenselike logic. Such a language permits a very natural axiomatisation of the iterative conception, which upholds the Maximality thesis. It is argued that the modal approach is consonant with mathematical practice and a plausible metaphysics of sets and shown that MST interprets a natural extension of Zermelo set theory less the axiom of Infinity and, when extended with a further axiom concerning the extent of the hierarchy, interprets Zermelo-Fraenkel set theory. 1 The iterative conception. The iterative conception of set, which came to philosophical prominence through George Boolos’s famous paper, has it that all (pure) sets will be formed in the following way. At the very first stage, there are no sets, and none are formed. At each subsequent stage, all sets of sets formed at earlier stages will be formed. At the second stage, H will be formed; at the third, H and tHu will be formed; at the fourth, H, tHu, ttHuu and tH, tHuu will be formed; similarly for the fifth, the sixth, and so on. Immediately after these stages will come the first limit stage, when all sets of sets formed at a finite stage will be formed. At this stage, all the hereditarily finite sets will be formed. At the next stage, the first sets of infinite rank will be formed. And so it goes on, stretching as high as the ordinals are long. 1Boolos (1971). 2Throughout ‘set’ means pure set.