LINES OF RESEARCH:

Our lines of research can be grouped under three headings describing three stages in which we will tackle inexpressibility and incompleteness phenomena in the formal sciences.

Frameworks for studying Incompleteness and Reflection:

    The incompleteness phenomena, first studied by Goedel, impose limitations to the expressive and reflective power of formal systems for theories in the formal sciences. Because most research on incompleteness has been driven mainly by mathematical concerns, the chosen framework for studying these phenomena has usually been some lean mathematical theory such as Peano Arithmetic. However, while the official study of incompleteness is developed within a modest formal theory, much philosophical reflection on incompleteness takes place within a richer, often not explicitly articulated framework where one can, for instance, distinguish between formulae and their codes. Similarly, logicians are careful to make a distinction between the consistency of a system or the statement that a certain arithmetical sentence is not provable in the system, on the one hand, and the arithmetical sentence that is somehow associated with these non-provability claims, on the other. They are also mindful of the difficulties involved in articulating what it means for an arithmetical sentence to express a metalinguistic claim. While for many mathematical purposes it is not necessary to make these distinctions explicit in a formal theory, we expect that a formal treatment of these distinctions within appropriate formal theories can clarify some formal as well as philosophical issues. In the project we will devise such formal systems and analyse them.

    Philosophical reflection on incompleteness is often bound by restrictions of the formal framework in which the formal theories are developed. For instance, consider the study of proof theoretic reflection principles expressing the soundness of mathematical theories. The most direct expression of a soundness principle for a formal system is the statement that all of its theorems are true. But since a truth predicate is not generally available in the formal frameworks under consideration, the usual proof-theoretic reflection principles, which are formulated in these frameworks, are at most surrogates for the intended claim, which could be expressed more adequately by means of a global reflection principle that requires a truth predicate for its formulation. We will look at the question of which frameworks lend themselves to expressing soundness claims and related principles. In particular, we will review axiomatic theories of truth for their suitability in this respect and discuss the setting for the formulation of truth theories.

    In the study of truth, logicians tend to develop theories of truth for the language of arithmetic within arithmetic via a coding. As in the case of provability, however, this is in tension with informal practice in which they distinguish carefully between metatheoretic statements such as the truth of the theorems of an arithmetical system, on the one hand, and the sentences that are supposed to express such claims within their formal framework, on the other. We will consider truth theories in which the object language to which the truth predicate is applied is distinct from the language that is used to talk about truth bearers. The separation between the object-language and the language of the metatheory is relevant in the case of truth where the ontological commitments of the metatheory turn out to be relevant for the evaluation of incompleteness. It matters, for instance, whether truth is applied to codes of sentences, types of sentences, or propositions, and one's choices may well impinge on one's stance towards the phenomena under study.

    Overall, our aim will be to make explicit distinctions and assumptions usually left implicit in the informal metatheory in the study of incompleteness and inexpressibility phenomena.

Overcoming Incompleteness and Inexpressibility:

    One problem caused by the incompleteness phenomenon concerns the question of what it is to accept a mathematical theory. It is often argued that the acceptance of a theory outstrips the acceptance of its mathematical content. Our acceptance of Peano Arithmetic, for example, commits us not only to its theorems but also to its own consistency, even though, on a familiar interpretation of Goedel's second incompleteness theorem, the consistency of Peano Arithmetic could not possibly be a theorem of Peano Arithmetic. In fact, the acceptance of Peano Arithmetic may even commit us to stronger claims such as the statement that all of its axioms and theorems are true, or even the further claim that this very statement of soundness is true.

    We will build on work by Solomon Feferman and others in our attempt to make explicit what is implicit in the acceptance of a theory. We will compare various attempts in this direction. In particular we will investigate truth theories for theories in which these incompleteness phenomena occur, as they can be used to make explicit assumptions implicit in the acceptance of the original theories. Since in light of the results  of stage 1, we will be working in frameworks deviating from more common approaches, we expect that our work will shed new light on the role of axiom schemata and their indefinite extendibility.

    The open-ended nature of the universe of set theory brings out a different source of inexpressibility. Some view the open-ended nature of the universe as a result of what they call its indefinite extensibility. They think that there is no comprehensive domain of sets and, as a result, no absolutely general set-theoretic statements. A challenge for them is to find other means to do justice to the intended generality of set theory. Since set theory is often taken to provide the universe of mathematics, the problem afflicts all the other formal sciences. One option is to read their principles as implicitly modalised claims, which hold no matter how we extend the range of our quantifiers.  To carry out this programme, we need both to be able to systematically translate axiomatic theories into suitably modalised theories and provide a satisfactory account of the modality. Of particular interest is the case of set theory, as one can exploit the modal operators to give an account of the iterative conception that takes the tense in the informal metaphor of set formation seriously.

    Set theory is generally taken to provide the ontology for the model-theory of formal languages. Logical consequence is defined in terms of truth in a set-theoretic structure. As a result, model-theoretic theorising for certain formal languages faces distinctive difficulties. The omission of a comprehensive domain is of little consequence for the model-theory of first-order languages, which we know to be extensionally adequate in virtue of the completeness of the predicate calculus. However, we cannot take refuge in this observation when we ascend to higher-order languages, where extensional adequacy turns on set-theoretic reflection principles of varying strength. We expect our later work on extensions of set- theoretic reflection to bear on this issue.

    There is a more general problem: no model theory that treats structures as objects can capture every legitimate interpretation of the language. One way to deal with this limitative result is to treat structures as values of higher-order quantifiers. A model theory for an nth-order language will thus take place in an n+1th language and the result will be an open-ended hierarchy of higher-order languages. An alternative for first-order languages enriched with absolutely general quantifiers could be to work in a first-order metalanguage expanded with a truth predicate and give a model theory that avoids appeal to set-theoretic structures. We know, after all, that in some contexts, we can use a truth predicate to simulate second-order quantification. The question is whether such an approach can deliver an adequate model-theory in line with a satisfactory account of logical consequence.

    Thus, while incompleteness phenomena are generally studied in isolation from the problem of absolute generality, we think work on schemata, modal devices and truth predicates will prove helpful in both areas.

Open-endedness and Inexpressibility:

    Inexpressibility can arise in connection to deductive incompleteness, on the one hand, and to what we may call ontological incompleteness, on the other.

    In particular, we will focus on expressive limitations caused by the open-ended nature of the mathematical universe, which provides a domain for all the formal sciences. It is not clear that our set- theoretic theorizing can characterise the universe of sets. Instead, we have an unending sequence of ever more comprehensive domains, all of which appear to do justice to the axioms of set theory even when these are suitably supplemented with various large cardinal hypotheses. While some theorists view the open-ended character of the universe to be evidence against the absolute generality of the formal sciences, others take the open-ended nature of the universe to be no obstacle for a comprehensive domain of sets. One challenge for the latter is to articulate the open-ended nature of the universe in different terms. One option would be to say that the extent of the universe places serious limits on our ability to characterise the domain of all sets. Any putative characterisation of the domain fails to distinguish it from a less-than-comprehensive domain, indeed, a set-sized one. The content of this inchoate thought is, however, very sensitive to the expressive resources of our language. If we restrict ourselves to a first-order language, then the claim becomes a theorem of Zermelo-Fraenkel set theory. In a second-order language, the thought becomes a relatively modest large cardinal hypothesis. However, as shown by William Reinhardt, the thought becomes inconsistent in a third-order language with third-order parameters. We will explore the limits of reflection with an eye to clarifying its potential foundational role.

    To express these reflection principles, we will also consider extensions of the language of set theory by a truth predicate. Unlike in previous cases, we would, for example, be able to formulate a principle of global set-theoretic reflection whereby for any set of true sentences of the language, there is a set M such that every sentence of the language of set theory in the set is true when suitably relativised to M. There is, in particular, a set, which makes true relativisations of all true sentences of the initial language of set theory and is therefore completely indistinguishable in the language from the entire universe. Matters become more complicated when we allow the truth predicate in the sentences we reflect down to a set. Another reason to be interested in such reflection principles has to do with Feferman's suggestion to employ reflection principles to provide a set-theoretic characterisation of the reflective closure of certain set theories.

    There is another respect in which open-endedness of the set-theoretic universe can impinge on the development of a theory of truth. Often, a theory of truth builds a transfinite hierarchy of truth predicates on a given base theory. But while these truth predicates generally presuppose a fixed array of ordinals to serve as indices for the truth predicates in the hierarchy, we should probably think of the ordinals themselves as forming an open-ended series. One alternative would be to index the truth predicates not by ordinals but rather by codes of formulae in the language of set  theory which provably define unique set-theoretical ordinals. By altering what is meant by `provably' in this context one may capture a variety of hierarchies from those of a clear fixed height to autonomous progressions. The limits of such hierarchies have connections, not just with pure set theories,  but with questions on the limits of predicativity.