The Iterative Conception of Set

J Philos Logic (2013) 42: 697. doi:10.1007/s10992-012-9245-3

The final publication is available at

Abstract: 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.