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• W2625922173 abstract "The study of set theory (a mathematical theory of infinite collections) has garnereda great deal of philosophical interest since its development. There are several reasonsfor this, not least because it has a deep foundational role in mathematics; anymathematical statement (with the possible exception of a few controversial examples)can be rendered in set-theoretic terms. However, the fruitfulness of set theoryhas been tempered by two difficult yet intriguing philosophical problems: (1.) thesusceptibility of naive formulations of sets to contradiction, and (2.) the inability ofwidely accepted set-theoretic axiomatisations to settle many natural questions. Bothdifficulties have lead scholars to question whether there is a single, maximal Universeof sets in which all set-theoretic statements are determinately true or false (often denotedby ‘V ’). This thesis illuminates this discussion by showing just what is possibleon the ‘one Universe’ view. In particular, we show that there are deep relationshipsbetween responses to (1.) and the possible tools that can be used in resolving (2.).We argue that an interpretation of extensions of V is desirable for addressing (2.) ina fruitful manner. We then provide critical appraisal of extant philosophical viewsconcerning (1.) and (2.), before motivating a strong mathematical system (knownas‘Morse-Kelley’ class theory or ‘MK’). Finally we use MK to provide a coding ofdiscourse involving extensions of V , and argue that it is philosophically virtuous. Inmore detail, our strategy is as follows:Chapter I (‘Introduction’) outlines some reasons to be interested in set theoryfrom both a philosophical and mathematical perspective. In particular, we describethe current widely accepted conception of set (the ‘Iterative Conception’) on whichsets are formed successively in stages, and remark that set-theoretic questions canbe resolved on the basis of two dimensions: (i) how ‘high’ V is (i.e. how far we goin forming stages), and (ii) how ‘wide’ V is (i.e. what sets are formed at successorstages). We also provide a very coarse-grained characterisation of the set-theoreticparadoxes and remark that extensions of universes in both height and width are relevantfor our understanding of (1.) and (2.). We then present the different motivationsfor holding either a ‘one Universe’ or ‘many universes’ view of the subject matter ofset theory, and argue that there is a stalemate in the dialectic. Instead we advocatefilling out each view in its own terms, and adopt the ‘one Universe’ view for thethesis.Chapter II (‘G¨odel’s Programme’) then explains the Universist project for formulatingand justifying new axioms concerning V . We argue that extensions of V arerelevant to both aspects of G¨odel’s Programme for resolving independence. We alsoidentify a ‘Hilbertian Challenge’ to explain how we should interpret extensions ofV , given that we wish to use discourse that makes apparent reference to such nonexistentobjects.Chapter III (‘Problematic Principles’) then lends some mathematical precisionto the coarse-grained outline of Chapter I, examining mathematical discourse thatseems to require talk of extensions of V .Chapter IV (‘Climbing above V ?’), examines some possible interpretations ofheight extensions of V . We argue that several such accounts are philosophicallyproblematic. However, we point out that these difficulties highlight two constraintson resolution of the Hilbertian Challenge: (i) a Foundational Constraint that we donot appeal to entities not representable using sets from V , and (ii) an OntologicalConstraint to interpret extensions of V in such a way that they are clearly differentfrom ordinary sets.5Chapter V (‘Broadening V ’s Horizons?’), considers interpretations of width extensions.Again, we argue that many of the extant methods for interpreting this kindof extension face difficulties. Again, however, we point out that a constraint is highlighted;a Methodological Constraint to interpret extensions of V in a manner thatmakes sense of our naive thinking concerning extensions, and links this thought totruth in V . We also note that there is an apparent tension between the three constraints.Chapter VI (‘A Theory of Classes’) changes tack, and provides a positive characterisationof apparently problematic ‘proper classes’ through the use of plural quantification.It is argued that such a characterisation of proper class discourse performswell with respect to the three constraints, and motivates the use of a relatively strongclass theory (namely MK).Chapter VII (‘V -logic and Resolution’) then puts MK to work in interpretingextensions of V . We first expand our logical resources to a system called V -logic,and show how discourse concerning extensions can be thereby represented. We thenshow how to code the required amount of V -logic usingMK. Finally, we argue thatsuch an interpretation performs well with respect to the three constraints.Chapter VIII (‘Conclusions’) reviews the thesis and makes some points regardingthe exact dialectical situation. We argue that there are many different philosophicallessons that one might take from the thesis, and are clear that we do not commitourselves to any one such conclusion. We finally provide some open questions andindicate directions for future research, remarking that the thesis opens the way fornew and exciting philosophical and mathematical discussion." @default.
• W2625922173 created "2017-06-23" @default.
• W2625922173 creator A5050299288 @default.
• W2625922173 date "2017-02-28" @default.
• W2625922173 modified "2023-09-25" @default.
• W2625922173 title "Executing Gödel’s programme in set theory" @default.
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• W2625922173 hasPublicationYear "2017" @default.
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