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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Some 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.
I 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.
The following definitions are relevant to what follows. A formula is \(\Sigma _0\), \(\Pi _0\) or \(\Delta _0\) iff all its quantifiers are bounded. A formula is \(\Sigma _{n + 1}\) (resp. \(\Pi _{n + 1}\)) iff it is of the form \(\exists x [\phi (x)]\) (resp. \(\forall x [\phi (x)]\)) where \(\phi \) is \(\Pi _n\) (resp. \(\Sigma _n\)). A formula \(\chi \) is \(\Delta _n^T\) iff there are \(\Sigma _n\) and \(\Pi _n\) formulas \(\phi \), \(\psi \) such that \(T \vdash \chi \leftrightarrow \phi \) and \(T \vdash \chi \leftrightarrow \psi \). The superscripted T indicates all \(\Delta _n\) formulas are relative to a theory T for \(n > 0\).
\(\Sigma _1\) Collection is the following schema: \(\forall x \exists y [ \phi (x, y)] \rightarrow \forall a \exists b \forall x \in a \exists y \in b[\phi (x, y)]\), where \(\phi \) must be a \(\Sigma _1\) formula.
Feferman actually denotes the system \({\mathsf {FSC}}\uparrow \) for the theory in all finite types, and \({\mathsf {FSC}}\) for an intermediary system having only 2 types and unrestricted quantifiers.
I assume \(\in \)-induction rather than foundation to hold “officially”, so to speak, in \({\mathsf {iKP}}(\omega )\); foundation is demonstrably its contrapositive, given classical logic, so this makes little difference.
In intuitionistic set theories, ordinals are hereditarily transitive sets; this condition is readily seen to hold of \(\alpha \).
Specifically, I have in mind a formal claim to the effect that every non-empty set has an \(\in \)-minimal element.
It might seem strange that full replacement is justified, but only partial separation. There are two things to note here: first, it is important to note the conditional nature of replacement as opposed the straightforwardly existential form of separation. Whereas separation categorically asserts the existence of the separation set, in replacement we are only told that a replacement set exists under certain conditions, which are intuitively strengthened by the intuitionistic background logic for higher complexity formulas. I discuss these issues further in the next paragraph in the context of choice; many of the same considerations carry over. Also, it is worth noting that adding separation for higher complexity classes yields more classical logic, given choice, via a modification of the argument for Theorem 2.3. Thus justifying full separation would amount to justifying classical logic for all formulas. This is not the case for replacement.
The interpretation is one of the main results of Feferman (2010). \({\mathsf {FSC}}\) (which Feferman actually denotes \({\mathsf {FSC}} \uparrow \)) stands in much the same relation to \({\mathsf {SCS}}\) as Gödel’s theory T (employed in the “Dialectica” interpretation of Gödel 1958) does to Primitive Recursive Arithmetic. For more on the latter see Avigad and Feferman (1998).
Feferman himself and other predicativists take the axiom of infinity as self-evident; since \(\omega \) is inductively defined, it is crystal clear, indeed the canonical instance of a clear or definite totality. I am sceptical that appeals to clarity of this kind are very helpful, and this is why I take it that the axiom is as much in need of justification as other set existence axioms, like the powerset axiom.
Feferman actually uses the term ‘definite’ rather than ‘determinate’; I have opted for the latter to avoid confusion with ‘definite’ in the sense of indefinite extensibility, to be discussed later.
A parallel discussion along Dummettian lines, interpreting ‘indefinite’ as ‘indefinitely extendible’, might also be given; very similar conclusions would apply to Feferman’s motivation so construed to those we draw at the end of this section.
The structure behind \({\mathcal {P}}(\omega )\), for instance, would be \(\langle \omega , {\mathcal {P}}(\omega ), \in \rangle \).
See the appendix to Feferman (2014).
On this connection, for instance, Feferman writes that “truth in full [bivalence] is only applicable to perfectly clear structures” (Feferman 2011, 11); given the definition of definiteness just considered, this entails that a totality is definite only if its associated conceptual structure is clear. In the first passage of this section, Feferman suggests that clarity of a structure entails the definiteness of its associated totality, where he says that the clarity of \(\langle {\omega , Sc, <, 0, +, \times \rangle }\) entails the definiteness of \(\omega \). Putting these together, we arrive at the hypothesis that Feferman holds a structure to be clear if and only if its associated totality is definite.
Not all of them share the obvious metamathematical nature of Gödel’s theorems, as examples like the Paris-Harrington thereom show. Still, there is little doubt about the truth-values of any of these results in the standard model, in stark contrast to those examples condsidered below.
Here we think of \(\omega \) as the domain of \(\mathbb {N}\), the standard model of arithmetic.
cf. Kreisel (1967).
see e.g. Barwise (1975, 10).
However, \(\Pi _1^1\) formulas (in the analytical hierarchy) are absolute for standard transitive models of \({\mathsf {ZFC}}\); this shows some quantificational statements bounded to \({\mathcal {P}}(\omega )\) do not violate DET, and this will prove very important in the sequel.
See e.g. Kunen (2013, 123–124) for proof.
Rathjen (2016b) presently has this result of \({\mathsf {SCS}}\) together with global choice. The status of the full principle seems at present unclear.
By \(\Delta _1\)-\({\mathsf {LEM}}\) and -\({\mathsf {MP}}\), I mean exactly the schemas given as semi-logical axioms (b) and (c) but with \(\phi \) extended to range over \({\mathsf {SCS}}\)-provable \(\Delta _1\) formulas.
In the context of \({\mathsf {FSC}}\), this means that \(\lnot \lnot \exists x f(x) = 0 \rightarrow \exists x f(x) = 0\).
That \({\mathsf {FSC}}\) proves a functional translation of \(\Delta _1\)-\({\mathsf {LEM}}\) is actually found in some slides for a talk Feferman gave in 2012; see his website for details. Also, and perhaps more importantly, there is an ambiguity in my use of \({\mathsf {FSC}}\). Here, I am using it to refer to the system in 2 types over V for the translations; this system is interpreted in full \({\mathsf {FSC}}\), which Feferman denotes \({\mathsf {FSC}}\uparrow \), by using a variant of the Gödel Dialectica interpretation. The latter theory is the one Feferman shows can be interpreted in \(\mathsf {OST}\) in Feferman (2010).
One can show, e.g., that all theorems of \(ACA_0\) are theorems of \({\mathsf {SCS}}\) without too much work; since arithmetical formulas are \(\Delta _0\), the arithmetical comprehension axiom is essentially just an instance of \(\Delta _0\)-\(\mathsf {Sep}\). The exact amount of second-order arithmetic that holds in various semi-constructive set theories is an interesting open question; further work here is needed.
Feferman himself seems to express a very similar sentiment here (Feferman 2014, 8), but the indeterminacy in the unrestricted all in the definition of powersets is not cashed out in terms of a failure of absoluteness.
It is worth remarking that I am here viewing \({\mathsf {SCS^+}}\) as a theory in the logical sense—a collection of sentences–rather than as a theory in the (non-algebraic) mathematical sense with an intended semantic interpretation. The reasons for this should emerge as we continue, but roughly I think that the theory in the mathematical sense with which we are concerned is \({\mathsf {ZFC}}\); only some concepts employed there are, in Feferman’s sense, vague, and the theory of \({\mathsf {SCS^+}}\) results from obviating vague concepts in deductions.
Basic properties of the constructible universe tell us there is an ordinal \(\varrho \) such that \(L[A]_{\varrho }\) contains \({\mathcal {P}}(\omega )^M\), A and the realizer for \(\lnot {\mathsf {CH}}\). All ordinals in B must be above \(\varrho \).
See Rumfitt (forthcoming) for an argument here, based on Kreisel’s (1967)
References
Arrigoni, T., & Friedman, S. (2013). The hyperuniverse program. Bulletin of Symbolic Logic, 19(1), 77–96.
Avigad, J., & Feferman, S. (1998). Gödel’s functional (dialectica) interpretation. In Hanbook of proof theory (pp. 337–407). Elsevier Science B.V.
Barwise, J. (1975). Admissible sets and structures: an approach to definability theory. Berlin: Springer.
Devlin, K. (1993). The joy of sets. Berlin: Springer.
Diaconescu, M. (1975). The axiom of choice and complementation. Proceedings of the American Mathematical Society, 51, 171–178.
Feferman, S. (2009). Intellectica. Conceptions of the continuum, 51, 169–189.
Feferman, S. (2010). On the strength of some semi-constructive set theories. In Proof categories, and computation: Essays in honor of Grigori Mints (pp. 109–129). College Publications.
Feferman, S. (2011). Is the continuum hypothesis a definite mathematical problem? Part of the Exploring the Frontiers of Incompleteness Project: Available on Feferman’s website.
Feferman, S. (2014). Logic, mathematics and conceptual structuralism. In The metaphysics of logic (pp. 72–92). Cambridge University Press.
Gödel, K. (1958). Über eine bisher noch nicht benützte erweiterung des finitischen standpunktes. Dialectica, 12, 280–287.
Goodman, N., & Myhill, J. (1975). Choice implies excluded middle. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 24.
Koellner, P. (2006). On the question of absolute undecidability. Philosophia Mathematica, 14(2), 153–188.
Koellner, P. (2011). Feferman on the indefiniteness of CH. Part of the Exploring the Frontiers of Incompleteness Project.
Kreisel, G. (1967). Informal rigour and completeness proofs. Problems in the philosophy of mathematics, 47, 138–186.
Kunen, K. (2013). Set theory: An introduction to independence proofs (2nd ed.). London: College Publications.
Martin, D. (2001). Multiple universes of sets and indeterminate truth values. Topoi, 20(1), 5–16.
Meadows, T. (2013). What can a categoricity theorem tell us? Review of Symbolic Logic, 6(3), 524–544.
Rathjen, M. (2016a). Indefiniteness in semi-intuitionistic set theories: On a conjecture of Feferman. Journal of Symbolic Logic, 81(2), 742–754.
Rathjen, M. (2016b). On Feferman’s second conjecture (unpublished manuscript).
Rumfitt, I. (2015). The boundary stones of thought. Oxford: Oxford University Press.
Rumfitt, I. (forthcoming). Determinacy and bivalence. In The oxford handbook of truth. Oxford University Press.
Shapiro, S. (1990). Foundations without foundationalism. Oxford: Oxford University Press.
Väänänen, J. (2014). Multiverse set theory and absolutely undecidable propositions. In Interpreting Gödel. Cambridge University Press.
Weston, T. (1976). Kreisel, the continuum hypothesis, and second-order set theory. Journal of Philosophical Logic, 5(2), 281–298.
Woodin, H. (2001). The continuum hypothesis, part I. Notices of the American Mathematical Society, 48(6), 567–576.
Acknowledgements
I would like to thank Walter Dean, Hartry Field, Nick Jones, Alex Paseau and Jonathan Payne for their criticisms and suggestions. Special mention is due to Neil Barton, Ian Rumfitt, and Scott Sturgeon; without their encouragement this paper wouldn’t exist, and without their comments it would not be a fraction as good.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Scambler, C. An indeterminate universe of sets. Synthese 197, 545–573 (2020). https://doi.org/10.1007/s11229-016-1297-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-016-1297-4