Restrictiveness Relative to Notions of Interpretation

Abstract

Restrictiveness relative to notions of interpretationIncurvati, L.; Löwe, B. Published in:Review of Symbolic Logic DOI:10.1017/S1755020316000058 Link to publication Citation for published version (APA):Incurvati, L., & Löwe, B. (2016). Restrictiveness relative to notions of interpretation. Review of Symbolic Logic, 9(2), 238-250. DOI: 10.1017/S1755020316000058 General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. Abstract. 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 (1988aMaddy ( ,1988bMaddy ( , 1998. Maddy (1998Maddy ( , 1997 gives a semi-formal account of restrictiveness by defining a corresponding formal notion based on a class of interpretations. In (Löwe, 2001(Löwe, , 2003, Maddy's notion of restrictiveness was discussed and the theory ZFG (i.e., 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 ca...

What Is a Restrictive Theory?

“…is not an -model, then it cannot define any -models including the -model t • s(M ) = M ; 9 It is provisional because, as we shall see, it does not accommodate generic extension. 10 The use of these axiom removes some surprising inequivalences. so suppose s(M ) is an -model.…”

“…Let our axioms for ZFC include set-induction, the well-ordering theorem and the collection schema. 10 Let ZFC \{Inf} be ZFC without the axiom of infinity. Let ZFC fin be ZFC \{Inf} plus the negation of the axiom of infinity.…”

“… 5 For some criticism of Maddy’s analysis see [8], and for some approaches to its repair see [10]. Our analysis of Maddy’s work will be more coarse-grained here.…”

“…This will be discussed in more detail in Section 3.2. 5 For some criticism of Maddy's analysis see [8], and for some approaches to its repair see [10].…”

Maximality Principles in the Hyperuniverse Programme

“…219-232. Recently, the alleged 'restrictiveness' of V = L has been challenged byHamkins (2014), and further discussed byIncurvati and Löwe (2016).8 Or any two other theories whose consistency strength is, at least, that of ZFC+LC's.…”