Context & Aims of the Network

Set theory is in the throes of a foundational crisis, the results of which may radically alter our understanding of the infinite and mathematics as a whole. In essence, the idea that there is a unique, so to speak, place in which all of mathematics occurs, has become increasingly controversial. There are a variety of reasons for this development, but a common thread among them is a growing acceptance of indeterminacy in the concept of set and in the foundations of mathematics more generally. This has resulted in the emergence of a number of divergent perspectives regarding set theory and how it should be practiced. In order to properly understand what is at stake in the ensuing debates a great amount of expertise is required in both philosophy and mathematics. The Set Theoretic Pluralism (STP) network will enable world experts from across these disciplines to gain a better understanding of what is required to solve these problems and to lay the groundwork in for a new generation of philosophers and set theorists to tackle them.

Over the course of the twentieth century, set theory has become the de facto foundation for mathematics. It plays this role in two ways. First, it provides an ontology of spaces and objects which can represent the subject matter of contemporary mathematics. Second, it provides a lever via which the problems of contemporary mathematics may be solved. On the first front, set theory has been a success. On the second, however, significant problems have emerged. The most dramatic example of this is the continuum hypothesis (CH). While the large cardinal programme initially appeared to promise a means of solving these kinds of problems, it is now well-known that CH is independent of anything we could foreseeably think of as a large cardinal assumption.
 
In the last few years and in response to these epistemic challenges a number of new perspectives on set theory have emerged which attempt to engage with these problems by avoiding the fixed ontology of the cumulative hierarchy and replacing it with a plurality of universes. For multiverse approaches, a problem like CH is treated as misleading way of asking which universe we happen to be working in. For example, Joel Hamkins has proposed that set theory should be construed in better faith with its practice. In accord with contemporary set theory's fascination with models, Hamkins suggests that the models themselves should be added its ontology (Hamkins, 2012). John Steel takes the impressive impact of the large cardinal programme on descriptive set theory and turns our ordinary understanding of sets on its head. Rather than thinking of set theory as describing some pre-existing structure in which mathematics can be seen to take place, we should rather see it as a congenial scaffolding through which further concrete mathematics can be interpreted (Steel, 2012). Finally, Friedman’s hyperuniverse programme attempts to combine features of both the universe and multiverse perspectives. By tracking first order properties of universes in multiverses constrained by natural principles, Friedman aims to discover new axiom candidates to characterise the universe of sets V. Väänänen uses his dependence logic, in particular the concept of team semantics, to make sense of the multiverse idea. His starting point is general first order logic with multiverse structures and he applies this to set theory.

Each of these pictures admits a kind of pluralistic ontology and indeterminacy into foundations. The move is controversial. Hugh Woodin has argued that the kind of generic multiverse offered by Steel reduces set theory to a species of formalism that betrays its Cantorian roots (Woodin, 2012). Moreover, Tony Martin has offered a naïve re-working of Zermelo's categoricity argument to claim that the indeterminacy revealed by CH is of a merely epistemic nature and thus, that the metaphysical re-imaginings of Hamkins and Steel are unwarranted (Martin, 2001; Zermelo, 1976). In a related vein, a criticism of the pluralist account of foundations is given by Väänänen in his comparison of the second order logic and set theory approaches (Väänänen, 2012).

Beyond the mathematical challenges involved in addressing these programmes, there are significant overlaps with recent work in mainstream analytic philosophy, particularly in metaphysics and philosophical logic. A key problem in metaontology is Putnam’s paradox, which is a generalisation of Skolem’s paradox to language and semantics at large. Using model theoretic techniques, it is argued that we are caught in a regress of theory augmentation whenever we seek to give a full account of the meaning of our expressions. Without such an account, we lose the ability to anchor our ontology to our language. A response emerges with Lewis and has been developed by Sider, Schaffer and Williams. They argue that there is a privileged language which carves nature at it joins and that this is the goal of our best theories. For multiverse debates, these approaches are particularly useful for the one-universe adherent. Related work by Kennedy (2013) suggests a pluralistic approach involving generalised constructibility and more widely the concept of "formalism freeness", and its dual, the concept of the entanglement of a semantically given object with its underlying formalism. On the other hand, there has also been recent work into the identification of substantive debates. Stemming from Carnap (1956) and Ryle (1954) – and emerging more recently with Thomason (2009), Chalmers (2011) and Sider (2011), it is argued that some metaphysical debates are merely verbal. Such debates are pointless as although the parties to the debate are in conflict nothing substantive hangs on the result. With multiverse debates, these approaches provide a means of arguing that some questions are meaningless.

With regard to philosophical logic, a significant amount of recent activity has been devoted to problems of indeterminacy; in particular, problems caused by vagueness and the liar paradox. A prominent response to these problems is known as supervaluation. Observing that indeterminacy results where there are different possibilities none of which is determined as correct, supervaluation tells us that the determinate propositions are those which are true regardless of which possibility we select. In the context of the multiverse, a proposition is meaningful if it is true in every universe. This, however, is just one of many different approaches to indeterminacy which include epistemicism, fuzzy logic, non-standard consequence relations and paraconsistency (Williamson 2008). It has been observed that any approach to indeterminacy developed in one area can be generalised into an analogous response in another. This raises interesting questions about the applicability of a wider variety of techniques in philosophical logic to the multiverse. 

This STP network will provide opportunities for set theorists and philosophers to communicate the problems they are facing and lay the groundwork for their successful solution.

References

Tatiana Arrigoni and Sy-David Friedman. The hyperuniverse program. Bulletin of Symbolic Logic, 9(1):77–96, 2013.

Rudolf Carnap. Empiricism, semantics and ontology. In Meaning and Necessity. University of Chicago Press, Chicago, 1956.

David J. Chalmers. Verbal disputes. Philosophical Review, 120(4):515–566, 2011. 

Victoria Gitman and Joel David Hamkins. A natural model of the multiverse axioms. Notre Dame Journal of Formal Logic, 51(4):475–484, 2010.

Joel Hamkins. The modal logic of forcing. Transactions of the American Mathematical Society, 360(4):1793–1817, 2007.

Joel Hamkins. Some second order set theory. In R. Ramanujan and S. Sarukkai, editors, Logic and its Applications: Lecture Notes in Computer Science, volume 5378, pages 36–50. Springer-Verlag, Heidelberg, 2009.

Joel David Hamkins. The set-theoretic multiverse. The Review of Symbolic Logic, 5:416–449, 2012.

Juliette Kennedy. On Formalism Freeness: Implementing Gödel's 1946 Princeton Bicentennial Lecture. The Bulletin of Symbolic Logic, 19(3):351–393, 2013.

Juliette Kennedy, Menachem Magidor, and Jouko Väänänen. Inner Models from Extended Logics. Isaac Newton Institute preprint series, 2016.

Øystein Linnebo. Pluralities and sets. Journal of Philosophy, 107(3), 2010.

Penelope Maddy. Believing the axioms. I. Journal of Symbolic Logic, 53(2):481–511, 1988.

Donald A. Martin. Multiple universes of sets and indeterminate truth values. Topoi, 20(1):5–16, 2001.

Toby Meadows. What does a categoricity theorem tell us? Review of Symbolic Logic, (3):524-544, 2013.

Toby Meadows. Naive infinitism. Notre Dame Journal of Formal Logic, forthcoming - 2013.

Gilbert Ryle. Dilemmas. Cambridge University Press, 1954.

Jonathan Schaffer. Is there a fundamental level? Noûs, 37(3):498–517, 2003.

Theodore Sider. Writing the Book of the World. Oxford University Press, 2011.

J. Stavi and J. Väänänen, Reflection principles for the continuum. Logic and Algebra, ed. Yi Zhang, pp. 59-84, Contemporary Mathematics, Vol 302, AMS, 2002.

Amie Thomasson. Answerable and unanswerable questions. In David Chalmers, David Manley, and Ryan Wasserman, editors, Metametaphysics: New Essays on the Foundations of Ontology. Oxford University Press, 2009.

W. Hugh Woodin. Set theory after Russell; the journey back to eden. In G. Link, editor, 100 Years of Russell’s Paradox. De Gruyter, 2004.

John R. Steel. Gödel’s program. forthcoming, 2012.

Jouko Väänänen. Multiverse set theory and absolutely undecidable propositions. in: J. Kennedy (Ed.): Interpreting Gödel, Cambridge University Press, 2014, 180-208.

Jouko Väänänen. Second order logic and foundations of mathematics. Bulletin of Symbolic Logic 7(4), 2001.

Jouko Väänänen. Second order logic or set theory? Bulletin of Symbolic Logic, 18(1):91–121, 2012.

Jouko Väänänen. Dependence Logic: A New Approach to Independence Friendly Logic. London Mathematical Society Student Texts (No. 70) Cambridge University Press, 2007.

W. Hugh Woodin. The realm of the infinite. In Michael Heller and W. Hugh Woodin, editors, Infinity: New Research Frontiers. Cambridge University Press, Cambridge, 2011a.

W. Hugh Woodin. The Continuum Hypothesis, the Generic Multiverse of Sets, and the ?-Conjecture. Cambridge University Press, 2012.

Timothy Williamson. Vagueness, volume 81. Routledge, 1994.

Ernst Zermelo. On boundary numbers and domains of sets: new investigations in the foundations of set theory. In From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Oxford University Press, 1976.
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