An Indeterminate Universe of Sets Chris Scambler December 28, 2016 Abstract In this paper, I develop a view on set-theoretic ontology I call Universe-Indeterminism, according to which there is a unique but indeterminate universe of sets. I argue that Solomon Feferman’s work on semi-constructive set theories can be adapted to this project, and develop a philosophical motivation for a semi-constructive set theory closely based on Feferman’s but tailored to the Universe-Indeterminist’s viewpoint. I also compare the emergent Universe-Indeterminist view to some more familiar views on set-theoretic ontology. 1 Introduction The axioms of standard set theories are partly motivated by their association with the iterative conception of sets, according to which all sets exist in a well-founded cumulative hierarchy which is itself not a set. This picture is often itself motivated by describing a kind of set con- struction process, the output of which is the aforementioned cumulative hierarchy. The hypo- thetical process would go like this. In the beginning, there is nothing whatever, except you (or some other set-producing demiurge). At the zeroth stage of the process, form all the sets you can; since you have nothing around, this set will have nothing in it and so be the empty set. At any further successor stage, form all possible sets of things you had when done last time. At a limit stage, simply collect together all your work at previous stages. Every sequence of successor stages has a limit, and every limit has a further successor. There is, therefore, no end to the set construction process; according to the iterative conception, to be a set just is to be formed by you (or the demiurge) at some stage of said process. This informal picture is suggestive, but imprecise; however, it is approximated formally in all standard set theories, in the the sense that in all such theories one can define V0 = ;; Vα = P(Vα); +1 S Vλ = α<λ Vα and then prove that every set is contained in Vα for some α. This formal definition captures the intuitive content of the informal picture painted above, and the fact that standard set theories 1 prove every set to be produced at some stage Vα represents a kind of formal endorsement of the informal picture. There is something of a reflective harmony between the informal conception described in the previous paragraph and standard axiomatic set theories; while the informal conception com- mands serious consideration in virtue of its formalizability in standard set theories, at the same time standard set theories can seem arbitrary and unmotivated without the conceptual founda- tion and intuitive model provided for them by the iterative conception. As Kant might have put it, without formally axiomatized set theories the iterative conception is empty; but without the iterative conception, formally axiomatized set theories are blind. In the later part of the twentieth century, it was discovered that there are a vast number of questions we can ask about cumulative hierarchy structures in the formal languages of standard set theories that those theories are demonstrably incapable of answering, if they are consistent. The Continuum Hypothesis (CH) and the Generalized Continuum Hypothesis (GCH) are par- ticularly simple examples: they concern the jump in cardinality that occurs from one stage of the cumulative hierarchy to the next. According to CH, |V!+1| is @1 and thus only one infinite cardinality greater than V!; according to GCH, this is true of every stage. As is well known, Gödel showed that there was a model of the axioms of ZFC – the constructible universe, L – in which GCH holds; later, Cohen showed that we could find models for the axioms of ZFC wherein jV!+1j is @α for pretty much any α you care to name, subject to some restrictions. The method of Easton forcing allows GCH to be violated in equally radical ways. The pervasive nature of the independence phenomenon raises hard philosophical problems concerning the relationship between standard set theories and their informal motivation in the iterative conception. The iterative conception tells us that at each successor stage of the set construction process, we form all possible sets of things that were available to us last time; on the face of it, there is only one way to do this, so one would think that problems like CH and GCH should have definite truth-values. But the results just discussed suggest that standard set theories at the very least cannot tell us how many sets are formed at each stage, and so cannot determine what those truth-values are; and if standard set theories, which make the informal conception precise, cannot tell us this, then what can? There are (broadly speaking)1 two kinds of response to this situation in the literature. Universe-Determinism: Standard set theories describe a unique universe of sets, the cumula- tive hierarchy. CH, GCH and all other set-theoretic propositions have unique truth-values, which are determined by the nature of that universe. Our present incapacity to solve such problems is a reflection of a substantive incompleteness in standard set theories. Multiverse-Pluralism: The independence phenomenon reveals that standard set theories in fact describe many non-isomorphic universes of sets. The initial thought that the cumu- lative hierarchy was unique up to isomorphism was erroneous; in fact there are many such structures, each a legitimate and independent object of set-theoretic study. Since the truth-values of GCH, CH and many other set-theoretic propositions vary from universe to universe, these propositions do not have unique truth-values. 1Some views such as the “hyperuniverse” view of (Arrigoni and Friedman, 2013) are difficult to place; however I think it fair to say that most currently held views fit roughly into one or other of these categories. 2 The main goal of this paper is to develop and defend a third position, which cuts across these two. I call it Universe-Indeterminism: Standard set theories describe a unique universe of sets, the cumu- lative hierarchy. However, the iterative conception and its formal approximations employ concepts that are inherently indeterminate. The indeterminacy arises in the idea that, at successor stages, we collect together all possible sets of things available to us at previ- ous stages; formally, it arises through acceptance of the powerset axiom and its use in the recursive definition of the cumulative hierarchy. As a result of this indeterminacy, the uni- verse of all sets does not determine truth-values for set-theoretic propositions including CH and GCH. Universe indeterminism might seem like an unlikely view. Many authors who give arguments to the effect that the universe of sets is unique up to isomorphism or even for the quasi-categoricity of set theory seem to take it that these results entail bivalence for corresponding classes of statements in the language of set theory; in the former case, all of them, in the latter case, for claims that can be bounded to particular ranks. These entailments are prima facie plausible, but not, I would suggest, so obvious as to not require argument – which they all too rarely receive.2 One of the motivating thoughts behind this paper is that the entailments might not hold. One might take the arguments for uniqueness or quasi-categoricity as evidence that our concept of set singles out a unique structure or some unique collection of structures, but nevertheless hold that the structure(s) thus engendered are inherently indeterminate and don’t settle all the questions one can ask about them in formal languages.3 This line might sound as daft as its negation sounded plausible, but I hope to show in this paper that in fact there is a viable line here that is worth considering. To do so, I will take as a starting point commitment to the existence of a unique universe of sets. Given this, I will develop an axiomatization for set-theoretic truth in such a universe in a partially non-classical logic that a Universe-Indeterminist could take to formally flesh out their claims that certain set-theoretic propositions are indeterminate, and that is philosophically motivated by their ontological position. I shall also show how the position differs in detail from all brands of Multiverse-Pluralism, and briefly discuss the relative merits of the view. 2 Feferman’s Semi-Constructive Set Theory The view that standard set theories deploy concepts that are inherently indeterminate has been championed by Solomon Feferman; in what follows, I will be heavily guided by his recent formal and philosophical work on semi-constructive set theories. Such theories, according to Feferman, provide a formal framework in which to assess the determinacy or otherwise of 2I have in mind (Martin, 2001) in the former, and (Kreisel, 1967) in the latter. There is an interesting recon- struction of an argument Kreisel might have given here in (Rumfitt, forthcoming) 3I will henceforth largely ignore issues of height indeterminacy; for simplicity’s sake, I will often talk as though I have reason to believe ‘the’ universe is of determinate ordinal height, but none of my arguments will turn on this. Everything I will say applies equally well to a Zermelo-style height potentialist, though the statements become slightly more complicated. 3 problems like CH; recent work by Michael Rathjen has raised the stakes by showing that CH _ :CH is not derivable in a certain semi-constructive set theory. These claims and results are striking, and worthy of close consideration; I also think, and will argue, that they are useful in the development of the Universe-Indeterminist position.