Abstract
Surveying and criticising attitudes towards the role and strength of the iterative conception of set—widely seen as the justificatory basis of Zermelo-Fraenkel set theory with Choice—this paper highlights a tension in both contemporary and historic accounts of the iterative conception’s justificatory role: on the one hand its advocates wish to claim that it justifies ZFC, but on the other hand they abstain from stating whether the preconditions for such justification exists. Expanding the number of axioms that the conception is standardly charged with failing to justify, in the forms of the Emptyset and Powerset, this paper aims to extend the critique that the iterative conception does not justify ZFC to its subsystems. Exploring the historic and contemporary relationship between the iterative conception and set theory, the paper then attempts to defuse strategies that avoid the problems of intrinsic justification by weakening the iterative conception’s role to the ‘motivational’ or ‘heuristic’ by showing that what the iterative conception has been seen to motivate has changed over time. It is suggested that the conjunction of these arguments seriously weakens the programme of reasoning on an ‘atheoretic’ notion of set.
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Notes
Which states that for any arbitrary sets X and Y, if for every x in X, x is also in Y, then \(X = Y\).
Which states that if A is a set, so is its image under a class function F[A]: every x in A can be replaced under its image under a class function, \(x'\).
Which states that the Cartesian Product of any collection of nonempty sets is nonempty.
It is worth commenting that actual infinity is nevertheless supplemental to the iterative conception. As will be touched on in Sect. 4.1.1, this gives grounds to view Infinity as underivable from the iterative conception alone.
Despite considering it self-evident, Zermelo argued that, due to its consequences, Choice was ‘necessary for science’ (1908a, p. 131).
Kanamori (2012) notes that while Replacement is normally shoved to the back of textbooks, this is more to do with applicability than epistemic trust.
Extrinsic justifications appeal to the utility of an axiom’s consequences for mathematics, rather than their intuitive plausibility. See Gödel (1947, p. 521) for a classic articulation, and Maddy (1997) for detailed exposition. See Tiles (1989/2004, p. 208) and Tait (2005, p. 283) for critiques of extrinsic justification in set theory.
ZFC set theory—Extensionality, Replacement, and Choice.
I would like to thank an anonymous reviewer for asking for expansion and clarification of this section.
One might be sceptical that one can reasonably accept first-order Reflection but reject generalising upwards. The issues surrounding this are dealt with in the following section.
This is perhaps because the assumption of actual infinity, while fundamental to modern mathematics, is an auxiliary assumption that is brought to the iterative conception.
Boolos in his classic paper stops iteration at the first non-recursive ordinal, and in an introduction to Gödel’s writings on the iterative conception describes the iterative conception as a ‘general procedure’ in which ‘new operations’ arise out of set formation (Rin, 2015).
It might be instructive that this is how popular textbooks also see things: Enderton (1977) takes it as necessary for Replacement to guarantee sets beyond \(\omega \), and thereby transfinite recursion.
Martin (1970, p. 112) can also be seen as holding a form of this position. Despite advocating the iterative conception, he writes that, in addition to not being able to tell us ‘[h]ow long the iterative process continue[s]’, that it cannot tell us ‘[w]hat is meant by a “subset” either. This argument is also noted by Feferman (2000), with regards circularity, and Boolos again appears sceptical: in an argument separate from the iterative conception he writes that ‘it does not seem to me unreasonable to think that perhaps it is not the case that for every set, there is a set of all its subsets’ (2000, p. 267).
It is of note that Barton entertains working within ZFC—the Power Set.
Of which the derivation is trivial, as it is the assumption on which the iterative conception is based anyway.
This is the most generous reading of the iterative conception: many, including its advocates, note that it was only in the 1960s/70s that the iterative conception as we recognise it was formulated, as a derivation of the axioms from a stage theory (see Arrigoni, 2007, pp. 57–58, fn. 20; Barton, 2020, pp. 165–166; Kanamori, 2012, p. 77) An alternative critical reading, however, can be found in Morris (2018), who argues that Zermelo was developing a new concept of set when he devised the axioms, but that-contrary to the iterative conception’s claims for itself-that it was not a natural consequence of reflection on our concept, but an ad-hoc response to paradox (pp. 47–49).
CH states that the cardinality of the powerset of the Integers is identical to that of the Continuum. Locally, it states that
$$\aleph _1 = 2^{\aleph _0}$$Generally, it states that
$$\aleph _{\alpha + 1} = 2^{\aleph _\alpha }$$for any infinite cardinal.
Forcing simulates the addition of new objects into V to create larger models of ZFC. If taken as an axiom, as below, these are taken to already exist in V (Schatz, 2019, p. 13).
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Acknowledgements
I would like to thank both Laurenz Hudetz and Roman Frigg for their encouragement and support in my writing of this essay both as a master’s thesis and in its present form, as well as their encouragement in submitting this essay for publication. All errors contained within are my own. I would also like to thank two anonymous referees for feedback on this paper.
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Glasman, T. The iterative conception of set does not justify ZFC. Synthese 203, 36 (2024). https://doi.org/10.1007/s11229-023-04408-8
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DOI: https://doi.org/10.1007/s11229-023-04408-8