Abstract
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, I outline a different approach, which circumvents these problems by dispensing with the priority or dependence relation altogether. Along the way, I show how this approach enables the mathematical structuralist to defuse an objection recently raised against her view.
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Notes
In the remainder of this paper, I will dispense with the use of quotation marks to signal the distinction between use and mention, whenever this aids readability without jeopardizing understanding.
The diagnosis of the paradoxes offered here is indebted to that offered in Rieger (2009).
In technical terms, we say that a set is pure iff no individuals belong to its transitive closure.
The issue of which of the ZFC axioms are sanctioned by the iterative conception was famously discussed by Boolos (1971, 1989), who argued that Choice, Extensionality and Replacement do not follow from the conception. Now the axioms of Z do include Extensionality. However, Boolos’s point with regards to this axiom seems to be that our reasons for accepting it are that it serves to distinguish sets from other entities, rather than that it is sanctioned by a certain conception of set. This seems right, but it seems to apply to other conceptions of set as well, as Paseau (2007, 44–46) has, to my mind, convincingly argued. Having said that, it would perhaps be more correct to say that the axioms of Z are sanctioned by the iterative conception and the concept of set. For ease of exposition, however, I will drop the qualification in the remainder of this paper.
More precisely, Separation is the schema: \(\forall z \exists y \forall x (x \in y \leftrightarrow x \in z \wedge \phi(x)),\) where ϕ(x) is, again, any formula in \(\fancyscript{L}_{\in}\) in which x is free and which contains no free occurrences of y.
Set-theorists put the matter by saying that the formula \(y = \fancyscript{P}(x)\) is not absolute for transitive models, where a model \({\mathfrak{A}}\) is said to be transitive iff every element of \({\mathfrak{A}}\) which is not an individual is also a subset of \({\mathfrak{A}}.\) In set theory a formula ϕ with free variables \(x_1, \ldots, x_n\) is said to be absolute for a model \({\mathfrak{A}}\) iff \({(\forall x_1, \ldots, x_n \in \mathfrak{A}) (\phi(x_{1}, \ldots, x_{n}) \leftrightarrow \phi^{\mathfrak{A}}(x_{1}, \ldots, x_{n}))},\) where \(\phi^{\mathfrak{A}}\) is the relativization of ϕ to \({\mathfrak{A}}\) obtained in the usual manner (i.e. by replacing ∀x and ∃x by, respectively, \({\forall x \in \mathfrak{A}}\) and \({\exists x \in \mathfrak{A}}\)). Effectively, this means that if ϕ is absolute for transitive models, to ascertain whether the property expressed by ϕ holds of a set, it is enough to ascertain whether ϕ holds of that set in any transitive model (and similarly for the case of relations). This cannot be done in the case of \(y = \fancyscript{P}(x),\) since y may be the set of all subsets of x in a certain transitive model and yet not be its powerset in the cumulative hierarchy: we can never rule out that, when we consider the whole set-theoretic universe, subsets can become definable that were not definable before, thereby affecting the truth-value of \(y = \fancyscript{P}(x).\) See Hallett (1984, 206–207) and Jech (2003, 163, 185) for more on the notion of absoluteness.
‘We may also appeal to the reflection principles to argue that the unbounded quantifiers are not really unbounded’ (Wang 1974, 209).
That is: \(\forall v \exists!w \phi (v, w) \rightarrow \forall u \exists y \forall x (x \in y \leftrightarrow \exists v (v \in u \wedge \phi(v, x))\). What this says, effectively, is that the image of a set in a first-order specifiable function is a set.
That is, by defining ‘it is possible that p’ as ‘it is not necessary that not p.’
In fact, there are theories which, albeit consistent if ZF is, allow for sets of this kind. The most famous examples, perhaps, are the four theories described by Aczel (1988), all of which are obtained by replacing ZFC ’s Axiom of Foundation with an anti-foundation axiom.
Of course, as Lowe himself is eager to stress, endorsing the claim that everything depends upon itself does not commit one to the claim that everything depends solely upon itself.
To be sure, in the quoted passage Cameron is not dealing with the question whether the membership relation has certain structural properties. The point is just that, if someone wants to use the notion of metaphysical dependence to motivate certain structural features of the membership relation, they cannot rely on our understanding of the membership relation to motivate the structural features of the relation of metaphysical dependence.
Strictly speaking, what we are describing here is non-eliminative structuralism. Eliminative structuralism combines the insight that mathematical theories describe structures with an attempt to avoid commitment to an ontology of structures. See Linnebo (2008, 60–61) for more on the distinction, and for a careful description of different brands of non-eliminative structuralism.
See Linnebo (2008, 66–68) for evidence that Resnik too is committed to the dependence claim, albeit less explicitly.
Elsewhere, Gödel (1947, 180, fn, 13) also points out that albeit the expression ‘set of x’s’ cannot be defined away, it can be paraphrased by other expressions such as ‘combination of any number of x’s’, of which, he says, it is reasonable to claim that we have an independent grasp.
To be precise, this is Horwich’s view as expressed in the first edition of Horwich (1998). In response to criticisms, Horwich has slightly modified his view about what the minimal theory of truth should comprise. However, the point is irrelevant for our present concerns, since his reasons for taking something to be part of the minimal theory remain the same. For details, the reader can consult the postscript to the second edition of Horwich’s book.
Some arguments to this effect are offered in Fine (1994). For the record, I do not find these arguments very convincing, but, for reasons of space, I cannot discuss them here.
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Acknowledgments
Many thanks to Tim Bays, Tim Button, Mic Detlefsen, Hannes Leitgeb, Hugh Mellor, Alex Oliver, Michael Potter, Peter Smith, Rob Trueman, Sean Walsh and Nathan Wildman. Earlier versions of this material were presented at a meeting of the Cambridge Serious Metaphysics Group and at the Foundations of Mathematics Seminar, University of Paris 7—Diderot. I am grateful to the members of these audiences for their valuable feedback. Research for this paper was made possible by a Research Fellowship from Magdalene College, Cambridge. I gratefully acknowledge the support of this institution.
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Incurvati, L. How to be a minimalist about sets. Philos Stud 159, 69–87 (2012). https://doi.org/10.1007/s11098-010-9690-1
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DOI: https://doi.org/10.1007/s11098-010-9690-1