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Scuola Normale Superiore di Pisa CLASSE DI LETTERE Corso di Perfezionamento in Filosofia PHD degree in Philosophy Structural Reflection and the Ultimate L as the true, noumenal universe of mathematics Candidato: Relatore: Emanuele Gambetta Prof.- Massimo Mugnai anno accademico 2015-2016 Structural Reflection and the Ultimate L as the true noumenal universe of mathematics Emanuele Gambetta Contents 1. Introduction 7 Chapter 1. The Dream of Completeness 19 0.1. Preliminaries to this chapter 19 1. G¨odel'stheorems 20 1.1. Prerequisites to this section 20 1.2. Preliminaries to this section 21 1.3. Brief introduction to unprovable truths that are mathematically interesting 22 1.4. Unprovable mathematical statements that are mathematically interesting 27 1.5. Notions of computability, Turing's universe and Intuitionism 32 1.6. G¨odel'ssentences undecidable within PA 45 2. Transfinite Progressions 54 2.1. Preliminaries to this section 54 2.2. Gottlob Frege's definite descriptions and completeness 55 2.3. Transfinite progressions 59 3. Set theory 65 3.1. Preliminaries to this section 65 3.2. Prerequisites: ZFC axioms, ordinal and cardinal numbers 67 3.3. Reduction of all systems of numbers to the notion of set 71 3.4. The first large cardinal numbers and the Constructible universe L 76 3.5. Descriptive set theory, the axioms of determinacy and Luzin's problem formulated in second-order arithmetic 84 3 4 CONTENTS 3.6. The method of forcing and Paul Cohen's independence proof 95 3.7. Forcing Axioms, BPFA assumed as a phenomenal solution to the continuum hypothesis and a Kantian metaphysical distinction 103 3.8. Woodin's program applied to third-order arithmetic, Woodin's Maximum and a comparison between Turing's completeness and Woodin's completeness 111 Chapter 2. Reflection 127 0.1. Preliminaries to this chapter 127 1. The Reflection Principle 130 2. Indescribability 133 3. Koellner's limitative results about Tait's reflection principles 137 4. Welch's Global Reflection 140 Chapter 3. Structural reflection 143 0.1. Preliminaries to this chapter 143 1. Structural reflection and the philosophical concept of richness 144 2. Beginning inner model theory 149 3. Relativising structural reflection to inner models 153 4. The Core Model and Structural Reflection 157 5. The theory of 0y 161 6. The theory of L[E∗], extender models 167 7. The HOD conjecture and the Wholeness axioms 171 8. The Ultimate L model 175 9. The philosophy of mathematics that I sustain 187 Chapter 4. Philosophical Aspects 197 0.1. Preliminaries to this chapter 197 CONTENTS 5 1. Philosophy of the Infinite: comparison between Cantor's absolute infinite and the Absolute principle of Damascius and Plotinus. 198 2. Stressing philosophical aspects: the Kantian distinction again within set theory and the problematic nature of real numbers 207 3. The plausibility of a new axiom, namely ZFC + 0] exists 215 4. Melissus of Samo and Georg Cantor 222 5. John Duns Scotus, the infinite and philosophy of mind 225 6. Reinhardt Cardinals and Anselm's argument 230 7. Paradoxes and the Curry-Liar paradox 234 Chapter 5. Appendix 239 1. Silver indiscernibles 239 Bibliography 247 Bibliography 247 1. INTRODUCTION 7 1. Introduction In this dissertation I am going to discuss structural reflection and the philosophy of set theory. We can find many different kinds of reflection principles in set theory, but I found structural reflection, as conceived by Joan Bagaria, an interesting and powerful method to characterize large cardinals in terms of reflection. In fact, structural reflection can produce a proper class of supercompact cardinals and a proper class of extendible cardinals (by using different conditions). Thus, structural reflection is important because it provides an intrinsic philosophical justification of large cardinals. In fact, the large cardinals, that we are able to interpret as principles of structural reflection are fundamental for Ω-logic and second-order arithmetic. By adopting structural reflection, we reflect an internal structural property of the membership relation. We can try to clarify immediately what one may mean by reflecting an internal structural property of the membership relation by following Bagaria's thought. We could answer that it is a property of some structure of the form (X; 2; (Ri)i2I ), where X is a set or a proper class and (Ri)i2I is a family of relations on X, and where I is a set that may be empty. So, an internal structural property of 2 would be formally given by a formula φ(x), possibly with parameters, that defines a class of structures of the form (X; 2; (Ri)i2I ). We might interpret this fact by saying that there exists an ordinal α that reflects φ and such that for every structure A in the class (that is, for every structure A that satisfies φ) there exists a structure B also in the class which belongs to Vα and is like A. Since, in general, A may be much larger than any B in Vα, the closest resemblance of B to A will be attained in the case that B can be elementarily embedded into A. Thus we can now formulate the principle of structural reflection as follows: Definition 1. (Bagaria) (Structural reflection, SR) For every definable (in the first order language of set theory, with parameters) class of structures C of the form (X; 2 8 CONTENTS Vα ; (Ri)i2I ), there exists α such that α reflects C, i.e. C = C \ Vα and for every A in C there exists B in C \ Vα and an elementary embedding from B into A. From a mathematical perspective, the main objective of this dissertation is the ap- plication of structural reflection to the canonical inner model for a measurable cardinal, namely L[U]. following Bagaria's thought and his results [Bagaria 13], I will prove that structural reflection for Π1 classes of structures definable in V relativized to this canonical inner model is equivalent to the existence of 0y. Then, I will prove that structural reflection for classes of structures (whatever complexity) definable within L[U] is implied by the exis- y tence of 0 . The mathematical result concerning classes of structures Π1 definable in V will support my philosophical thesis that considers Woodin's Ultimate L as the true, noumenal universe of mathematics very close to V. In fact, I will prove that if we apply structural reflection with (Π1 definable in V) classes of structures to a weak extender model for a supercompact cardinal, we do not get transcendence over this inner model. At the same time, I will conjecture that if we apply structural reflection to a canonical inner model for a strong cardinal, we obtain 0{. Bagaria [Bagaria 13] proved that 0] existence is equivalent to structural reflection (with Π1 definable classes of structures in V) relativized to G¨odel's constructible universe L. Therefore, on one side, if we relativize structural reflection with Π1 definable classes of structures in V to G¨odel'sconstructible universe, inner model of iterated sharps, inner model of measurability, inner model of iterated daggers and inner model for a strong cardinal we obtain a sharp for these specific inner models. On the other side, if we relativize structural reflection for Π1 classes of structures definable in V to a weak extender model for a supercompact cardinal, we do not get transcendence. So, we may assert that structural reflection supports the philosophical thesis claiming that Woodin's Ultimate L (not yet constructed) is very close to V and it can be considered as the true, noumenal (I will clarify immediately the philosophical meaning of this word) uni- verse of mathematics where undecided mathematical statements, such as the Continuum 1. INTRODUCTION 9 Hypothesis, are settled. There are many aspects that justify structural reflection itself. First of all, Σ1 structural reflection can be proved from the axioms of ZFC. So, structural reflection is a feature of the universe of sets. Secondly, we have the concept of richness. When we relativize structural reflection to inner models, we are able to transcend the specific inner model and obtain a bigger, richer universe of sets. Thus, the philosophical concept of richness justifies structural reflection. On the one side, structural reflection pro- vides intrinsic philosophical justifications for large cardinals. On the other side, we have philosophical reasons that support structural reflection and render structural reflection a powerful method to interpret large cardinals as principles of reflection. Therefore, from a philosophical perspective, structural reflection is able to justify intrinsically large cardinal notions such as infinitely-many Woodin cardinals and a proper class of Woodin cardinals, that are fundamental for second-order arithmetic and Ω-logic. The precedent two cardinal notions reduce the phenomenon of incompleteness which arises within the universe of sets. Surely, infinitely-many Woodin cardinals and a proper class of Woodin are justified also extrinsically (fruitfullness of the results, i. e., what they are able to prove). Structural reflection is fundamental from a philosophical perspective because it gives us an intrinsic philosophical justification of large cardinals. Intrinsic philosophical justifications are based on the conceptual analysis of the sets themselves. In fact, as we will see, reflec- tion is an essential feature of the universe of sets. Bill Tait's reflection [Koellner 09] could not overcome the barrier represented by G¨odel'sconstructible universe and Philip Welch's [Welch 10] global reflection implies embeddings of proper classes which can be seen as problematic mathematical objects. Therefore, structural reflection seems to be more nat- ural and a more powerful method to characterize very large cardinal numbers. The best feature of this kind of reflection is that it seems to improve on Tait's and Welch's methods of reflection.
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