Restrictiveness Relative to Notions of Interpretation

Maddy gave a semi-formal account of restrictiveness by defining a formal notion based on a class of interpretations and explaining how to handle false positives and false negatives. Recently, Hamkins pointed out some structural issues with Maddy’s definition. We look at Maddy’s formal definitions from the point of view of an abstract interpretation relation. We consider various candidates for this interpretation relation, including one that is close to Maddy’s original notion, but fixes the issues raised by Hamkins. Our work brings to light additional structural issues that we also discuss. §1. Motivation There is a tradition of naturalistic philosophy of set theory in which philosophers observe that some theories are proposed by set theorists as reasonable contenders for foundations of mathematics and others are not. The naturalistic philosopher aims to understand the reasons for these decisions of the mathematical experts, preferably on the basis of mathematical understanding of the theories involved. The most prominent approach for this is to rule out theories because they restrict the development potential of the foundations of mathematics, i.e., violate the maxim maximize (Maddy, 1988a,b, 1998). Maddy (1998, 1997) gives a semi-formal account of restrictiveness by defining a corresponding formal notion based on a class of interpretations. In (Löwe, 2001, 2003), Maddy’s notion of restrictiveness was discussed and the theory ZF + ‘Every uncountable cardinal is singular’ was presented as a potential witness to the restrictiveness of ZFC. More recently, Hamkins has given more examples and pointed out some structural issues with Maddy’s definition (Hamkins, 2013). In this paper, we shall look at Maddy’s definitions from the point of view of an abstract interpretation relation. We shall then consider various candidates for this interpretation relation, including one that is close to Maddy’s original notion, but fixes the issues raised by Hamkins (2013). Our work brings to light additional structural issues that we also discuss. The first author acknowledges the support of Magdalene College, Cambridge during the time that this paper was written. The second author acknowledges the support of the Isaac Newton Institute for Mathematical Sciences during the programme Semantics & Syntax in 2012 and the hospitality of Corpus Christi College, Cambridge during his stay in Cambridge and subsequent visits. 2 L. Incurvati & B. Löwe §2. Interpretations and restrictiveness In the following, we let E be an arbitrary interpretation relation between first-order theories. Throughout, we shall be dealing with set theories, by which we mean finite extensions of a basic set theory BST. For all applications in this paper (with one exception), we let BST = ZF, so the word “set theories” refers to axioms systems in the language L∈ of the form ZF+A where A is an axiom (or, equivalently, a finite set of axioms). If T and S are set theories, we say that T extends S (S ⊆ T ) if the set of S-theorems is a subset of the set of T -theorems. If S = ZF+A and T = ZF+B, we write S∪T for ZF+A∧B. There is a rich body of literature on the space of theories with interpretations between them (de Bouvère, 1965; Ahlbrandt & Ziegler, 1986; Visser, 2006). An interpretation of T in S is a syntactic object describing for every model of S a subclass of that model that forms a model of T (for definitions, cf. §2.2. below). In this literature, it is common to have interpretation relations that are reflexive (the identity is considered an interpretation) and transitive. In this paper, we deal with relations that are neither, and therefore, we keep the discussion at a sufficient level of generality, and make explicit all assumptions about E in the statement of theorems. In this paper, we consider three properties of interpretation relations: reflexivity and transitivity, as mentioned; and then a property we call monotonicity : if S E T and R ⊇ T , then S E R. 2.1. Restrictiveness As observed, Maddy’s enterprise uses the maxim maximize in order to rule out theories as restrictive. Here, we think of a theory T as restrictive if it restricts the possible lines of development of set theory, i.e., there is some other consistent desirable axiom that is incompatible with T . The following formal sequence of definitions follows Maddy’s notions from (Maddy, 1997, §6), except that we keep the notion of interpretation completely open for now and allow theories extending BST (whereas Maddy only allows theories extending ZFC). As usual, we write T S for T E S and S 6E T . Clearly, this is irreflexive independently of what properties E has. Moreover, if E is transitive, then so is . Definition 1. Let S and T be set theories. 1. We say that T E-recaptures S if there is a consistent extension T ∗ of T such that S E T ∗. 2. We say that S weakly E-maximizes over T if T S and T does not E-recapture S. (In symbols: T <weak S.) 3. We say that S strongly E-maximizes over T if it weakly E-maximizes over T and S ∪ T is inconsistent. (In symbols: T <strong S.) 4. We say that T is (weakly / strongly) E-restrictive if there is a consistent set theory T ∗ that (weakly/strongly) E-maximizes over T . Maddy does not have a notion of weak maximization and the corresponding notion of weak restrictiveness. We shall discuss the relation between weak and strong maximization in §§5.. We now note two facts that will be important later in the paper, viz. that the transitivity of E induces the transitivity of weak and strong E-maximization. Proposition 2. Suppose that E is transitive. If R, S, and T are set theories and