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  1. George Boolos (1998). Must We Believe in Set Theory? In Richard Jeffrey (ed.), Logic, Logic, and Logic. Harvard University Press. 120-132.
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  2. George Boolos (1989). Iteration Again. Philosophical Topics 17 (2):5-21.
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  3. George Boolos (1971). The Iterative Conception of Set. Journal of Philosophy 68 (8):215-231.
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  4. Manuel Bremer (2010). Universality in Set Theories. Ontos.
    The book discusses the fate of universality and a universal set in several set theories.
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  5. Thomas Forster (2008). The Iterative Conception of Set. Review of Symbolic Logic 1 (1):97-110.
    The phrase ‘The iterative conception of sets’ conjures up a picture of a particular settheoretic universe – the cumulative hierarchy – and the constant conjunction of phrasewith-picture is so reliable that people tend to think that the cumulative hierarchy is all there is to the iterative conception of sets: if you conceive sets iteratively, then the result is the cumulative hierarchy. In this paper, I shall be arguing that this is a mistake: the iterative conception of set is a good (...)
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  6. Luca Incurvati (2014). The Graph Conception of Set. Journal of Philosophical Logic 43 (1):181-208.
    The non-well-founded set theories described by Aczel (1988) have received attention from category theorists and computer scientists, but have been largely ignored by philosophers. At the root of this neglect might lie the impression that these theories do not embody a conception of set, but are rather of mere technical interest. This paper attempts to dispel this impression. I present a conception of set which may be taken as lying behind a non-well-founded set theory. I argue that the axiom AFA (...)
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  7. Luca Incurvati (2012). How to Be a Minimalist About Sets. Philosophical Studies 159 (1):69-87.
    According to the iterative conception of set, sets can be arranged in a cumulative hierarchy divided into levels. But why should we think this to be the case? The standard answer in the philosophical literature is that sets are somehow constituted by their members. In the first part of the paper, I present a number of problems for this answer, paying special attention to the view that sets are metaphysically dependent upon their members. In the second part of the paper, (...)
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  8. Richard Jeffrey (ed.) (1998). Logic, Logic, and Logic. Harvard University Press.
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  9. Øystein Linnebo (2010). Pluralities and Sets. Journal of Philosophy 107 (3):144-164.
    Say that some things form a set just in case there is a set whose members are precisely the things in question. For instance, all the inhabitants of New York form a set. So do all the stars in the universe. And so do all the natural numbers. Under what conditions do some things form a set?
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  10. Øystein Linnebo (2007). Burgess on Plural Logic and Set Theory. Philosophia Mathematica 15 (1):79-93.
    John Burgess in a 2004 paper combined plural logic and a new version of the idea of limitation of size to give an elegant motivation of the axioms of ZFC set theory. His proposal is meant to improve on earlier work by Paul Bernays in two ways. I argue that both attempted improvements fail. I am grateful to Philip Welch, two anonymous referees, and especially Ignacio Jané for written comments on earlier versions of this paper, which have led to substantial (...)
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  11. Christopher Menzel (2014). Wide Sets, ZFCU, and the Iterative Conception. Journal of Philosophy 111 (2):57-83.
  12. Christopher Menzel (1986). On the Iterative Explanation of the Paradoxes. Philosophical Studies 49 (1):37 - 61.
    As the story goes, the source of the paradoxes of naive set theory lies in a conflation of two distinct conceptions of set: the so-called iterative, or mathematical, conception, and the Fregean, or logical, conception. While the latter conception is provably inconsistent, the former, as Godel notes, "has never led to any antinomy whatsoever". More important, the iterative conception explains the paradoxes by showing precisely where the Fregean conception goes wrong by enabling us to distinguish between sets and proper classes, (...)
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  13. M. D. Potter (1993). Iterative Set Theory. Philosophical Quarterly 44 (171):178-193.
    Discusses the metaphysics of the iterative conception of set.
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  14. Adam Rieger (2011). Paradox, ZF and the Axiom of Foundation. In D. DeVidi, M. Hallet & P. Clark (eds.), Logic, Mathematics, Philosophy, Vintage Enthusiasms: Essays in Honour of John L. Bell. Springer.
    This paper seeks to question the position of ZF as the dominant system of set theory, and in particular to examine whether there is any philosophical justification for the axiom of foundation. After some historical observations regarding Poincare and Russell, and the notions of circularity and hierarchy, the iterative conception of set is argued to be a semi-constructvist hybrid without philosophical coherence. ZF cannot be justified as necessary to avoid paradoxes, as axiomatizing a coherent notion of set, nor on pragmatic (...)
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  15. Mark F. Sharlow (2001). Broadening the Iterative Conception of Set. Notre Dame Journal of Formal Logic 42 (3):149-170.
    The iterative conception of set commonly is regarded as supporting the axioms of Zermelo-Fraenkel set theory (ZF). This paper presents a modified version of the iterative conception of set and explores the consequences of that modified version for set theory. The modified conception maintains most of the features of the iterative conception of set, but allows for some non-wellfounded sets. It is suggested that this modified iterative conception of set supports the axioms of Quine's set theory NF.
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  16. Mark F. Sharlow (1987). Proper Classes Via the Iterative Conception of Set. Journal of Symbolic Logic 52 (3):636-650.
    We describe a first-order theory of generalized sets intended to allow a similar treatment of sets and proper classes. The theory is motivated by the iterative conception of set. It has a ternary membership symbol interpreted as membership relative to a set-building step. Set and proper class are defined notions. We prove that sets and proper classes with a defined membership form an inner model of Bernays-Morse class theory. We extend ordinal and cardinal notions to generalized sets and prove ordinal (...)
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  17. J. P. Studd (2012). The Iterative Conception of Set: A (Bi-)Modal Axiomatisation. Journal of Philosophical Logic 42 (5):1-29.
    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 (...)
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  18. W. W. Tait (2003). Zermelo's Conception of Set Theory and Reflection Principles. In Matthias Schirn (ed.), The Philosophy of Mathematics Today. Clarendon Press.
  19. William Tait, Constructing Cardinals From Below.
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  20. Rafal Urbaniak, Nominalist Neologicism.
    The goal is to sketch a nominalist approach to mathematics which just like neologicism employs abstraction principles, but unlike neologicism is not committed to the idea that mathematical objects exist and does not insist that abstraction principles establish the reference of abstract terms. It is well-known that neologicism runs into certain philosophical problems and faces the technical difficulty of finding appropriate acceptability criteria for abstraction principles. I will argue that a modal and iterative nominalist approach to abstraction principles circumvents those (...)
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  21. Rafal Urbaniak (2010). Neologicist nominalism. Studia Logica 96 (2):149-173.
    The goal is to sketch a nominalist approach to mathematics which just like neologicism employs abstraction principles, but unlike neologicism is not committed to the idea that mathematical objects exist and does not insist that abstraction principles establish the reference of abstract terms. It is well-known that neologicism runs into certain philosophical problems and faces the technical difficulty of finding appropriate acceptability criteria for abstraction principles. I will argue that a modal and iterative nominalist approach to abstraction principles circumvents those (...)
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  22. Gabriel Uzquiano (forthcoming). Varieties of Indefinite Extensibility. Notre Dame Journal of Formal Logic.
    We look at two recent accounts of the indefinite extensibility of set, and compare them with a linguistic model of the indefinite extensibility. I suggest the linguistic model has much to recommend over extant accounts of the indefinite extensibility of set, and we defend it against three prima facie objections.
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  23. Gabriel Uzquiano (2002). Categoricity Theorems and Conceptions of Set. Journal of Philosophical Logic 31 (2):181-196.
    Two models of second-order ZFC need not be isomorphic to each other, but at least one is isomorphic to an initial segment of the other. The situation is subtler for impure set theory, but Vann McGee has recently proved a categoricity result for second-order ZFCU plus the axiom that the urelements form a set. Two models of this theory with the same universe of discourse need not be isomorphic to each other, but the pure sets of one are isomorphic to (...)
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