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Paradox and Foundation Zach Weber Submitted in Total Fulfilment of The Paradox and Foundation Zach Weber Submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy May 2009 School of Philosophy, Anthropology and Social Inquiry The University of Melbourne This is to certify that - the thesis comprises only my original work towards the PhD, - due acknowledgement has been made in the text to all other material used, - the thesis is less than 100,000 words in length. Preface Dialethic paraconsistency is an approach to formal and philosophical theories in which some but not all contradictions are true. Advancing that program, this thesis is about paradoxes and the foundations of mathematics, and is divided accordingly into two main parts. The first part concerns the history and philosophy of set theory from Cantor through the independence proofs, focusing on the set concept. A set is any col- lection of objects that is itself an object, with identity completely determined by membership. The set concept is called naive because it is inconsistent. I argue that the set concept is inherently and rightly paradoxical, because sets are both intensional and extensional objects: Sets are predicates in extension. All consistent characterizations of sets are either not explanatory or not coherent. To understand sets, we need to reason about them with an appropriate logic; paraconsistent naive set theory is situated as a continuation of the original foundational project. The second part produces a set theory deduced from an unrestricted compre- hension principle using the weak relevant logic DLQ, dialethic logic with quantifiers. I discuss some of the problems involved with embedding in DLQ, especially related to identity and substitution. Then I outline how the basic toolkit of standard set theory may be developed in this logic from the naive set concept alone, up through the theories of ordinal and cardinal numbers, including Cantor's theorem. The in- famous paradoxes are just theorems, as are the existence of objects like a universal set, unrestricted compliments, and the full set of ordinals. This furnishes the start of a purely paraconsistent foundation for mathematics, providing recapture of clas- sical theorems. It is further demonstrated that paraconsistent naive set theory is able to establish strong results such as the existence of large cardinals, the axiom of choice, and a decision on the generalized continuum hypothesis. The founda- tional reduction to sets, then, is both philosophically illuminating and technically rewarding. To conclude, some mathematical results are used to answer outstanding ques- tions about infinity and the transconsistent|demarcating the line between trans- finite and absolute infinity, in the same way that Dedekind demarcated the line between finite and infinite: by turning a paradox into a definition. 3 4 PREFACE The main original contribution of this piece is an operational artifact of para- consistent machinery. The elemental chapter 5 capturing the theories of ordinal and cardinal numbers is the central moment. A reader already convinced of the intrinsic interest of a robustly inconsistent paraconsistent set theory may wish to begin there. ∗ ∗ ∗ A note on names. To token that some contradictions are true, Routley and Meyer had been using variations of dialectic, until 1981 when Routley and Priest co-coined the neologism dialeth(e)ism, or two-way truth, inspired by a remark of Wittgenstein's. (DL, then, originally stood for dialectical logic.) According to first-hand accounts, Priest and Routley then forgot to agree how to spell the ism, and it has appeared since both with (in Priest) and without (in Routley) the extra `e'. I use the e-free form uniformly, both for aesthetic preference and in honor of Routley, who inaugurated dialethic set theory in his [Rou77]. Further, in the mid-eighties Routley changed his surname to Sylvan, and both names can be found intermittently distributed in the literature. As all the works I cite by him were published as Routley, that epithet is the only one used here. ∗ ∗ ∗ Thanks are due foremost to my supervisor, Graham Priest, who's own work and patient hours of discussion have been invaluable. Greg Restall co-supervised and offered constructive discussion and comments. Audiences at the Melbourne Logic Group and the University of Melbourne Postgraduate Colloquia have listened and contributed over the years. Similar thanks are due the Melbourne-Adelaide Logic Axis, the Melbourne Postgraduate Logic Conference 2006, the APPC 2004, and the AAP in Australia and New Zealand 2004 - 8. Ross Brady has gone above and beyond in checking details and suggesting strategies for improving the proofs. Anonymous referees for [Webng], which is incorporated here into chapter 4, also provided a great deal of help. My office mates, in particular Conrad Asmus, have been constant sources of consultation and challenge, as have been many of the postgrads here. And Vicki Macknight has been loving, listening, and patiently encouraging throughout|and read the manuscript, twice. This work was funded by an International Postgraduate Research Scholarship and a Melbourne International Research Scholarship. My thanks go to the Aus- tralian government and its taxpayers, who I am sure are eagerly awaiting the rise of inconsistent set theory. Melbourne, May 2009 Contents Preface 3 Introduction 7 1. The Set Concept 7 2. Definitions and Diagonals 11 3. Dialethism and Paraconsistency 13 4. A Place to Stand 16 Chapter 0. Into the Dialethic Fields 19 = 1. Paraconsistent Set Theory in C1 19 2. Dialectics 21 3. Inconsistent Sets 28 Part One | Paradox 31 Chapter 1. First Foundations|The Nineteenth Century 33 1. Introduction 33 2. Cantor: From Transfinite Numbers to Sets 35 3. Dedekind 46 4. Frege: From Logic to Number 48 5. Paradox and Prospect 51 Chapter 2. The Absolute 57 1. Introduction 57 2. Equivalents of the Axiom of Choice 59 3. The View From Below 64 4. The View from Above 68 5. Conclusion 71 Chapter 3. Foundations in the Twentieth Century 73 1. Introduction 73 2. On Method 76 3. Paradise Lost, Paradise Regained 77 4. Hierarchy 82 5 6 CONTENTS 5. Of Comprehension 94 6. Conclusion 97 Part Two|Foundation 101 Chapter 4. On A Logic for Naive Set Theory 103 1. Introduction 103 2. Logics of Formal Inconsistency 104 3. A Paraconsistent Logic 109 4. Relevant Identity 114 5. Restricted Quantification 118 6. An Iatrogenic Disorder 122 Chapter 5. Elements 127 1. Introduction 127 2. Logic 128 3. Axioms 132 4. Basics 133 5. ZF 136 6. Ordinals 139 7. Global Choice 148 8. Cardinals 151 Chapter 6. Reflection and Large Cardinals 159 1. Introduction 159 2. Reflection Theorems 162 3. Ultrafilters 165 4. Axioms of Infinity 166 5. The End of the Ordinals 171 6. Reflecting the Universe 174 7. Conclusion 176 Chapter 7. On the Transconsistent 177 1. Characterizing the Absolute 178 2. Characterizing the Transconsistent 181 3. Characterizing the Mathematics 184 Conclusion 195 Bibliography 201 Introduction 1. The Set Concept The concept of a set is simple to state: A set is any collection of objects that is itself an object, with its identity completely determined by its members. Many set theory textbooks open by claiming that `set' cannot be formally defined, because it is too primitive, but this isn't so; we've just had a fine definition. This is the naive set concept, and can be completely characterized in the language of first order logic by the principles of abstraction and extensionality, x 2 fz : Φ(z)g $ Φ(x); x = y $ (8z)(z 2 x $ z 2 y); respectively. These clauses fix the meanings of 2 and =, the only non-logical parts of the vocabulary of set theory; they look very much like analytic definitions of predication and identity. From abstraction immediately follows the principle of comprehension, (9y)(8x)(x 2 y $ Φ(x)) (Later we will state these officially, specifying a language.) The clauses underpin broader mathematical needs in a natural way: The comprehension principle pro- vides for existence, and the extensionality principle governs uniqueness, of objects. A set is the unique extension of an arbitrary predicate. The reason textbooks claim that there can be no definition of set is not that the concept is somehow opaque or ambiguous. The issue is that the set concept is formally inconsistent. The concept is not indeterminate, or underdetermined; it is overdetermined. I will argue that the inconsistency|a paradox, since it is a contradiction inside a tautology|is not accidental, but inherent to the concept. A dialethia is a contradiction following from true premises by valid inference rules, a true inconsistency. Dialethism is the thesis that some theories comprised of inconsistent sentences are still true; some of the sentences about sets are incon- sistent, and they are still true. Insofar as sets are among the most basic, primitive objects that can be described at all, the presence of contradiction in them is trans- fixing. 7 8 INTRODUCTION Roughly, a set is a multiplicity that also has unity, or, in more antique terms, a many that is also a one. How can a many also be a one? It sounds inconsistent for something to be both singular and plural|and it is. But this way of putting the paradox has a mysterious ring to it; happily, modern logic allows us to to apply sophisticated tools to old problems, to make the meaning of the phrase, and so its deep tension, precise. Generally, a system is extensional if coincidence of members, or of some other objects (e.g. truth values) obtained from a list is sufficient to establish an identity. Logicians then often call a system intensional just if it is not extensional. But an intension, and so an intensional system, has an independent meaning: Identity and other relations are intensionally determined by properties or predicates, by defini- tions.
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