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Topological Set Theories and Hyperuniverses Andreas Fackler

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Topological Set Theories and Hyperuniverses Andreas Fackler Topological Set Theories and Hyperuniverses Andreas Fackler Dissertation an der Fakultät für Mathematik, Informatik und Statistik der Ludwig–Maximilians–Universität München vorgelegt von Andreas Fackler am 15. Dezember 2011 Topological Set Theories and Hyperuniverses Andreas Fackler Dissertation an der Fakultät für Mathematik, Informatik und Statistik der Ludwig–Maximilians–Universität München vorgelegt von Andreas Fackler am 15. Dezember 2011 Erstgutachter: Prof. Dr. Hans-Dieter Donder Zweitgutachter: Prof. Dr. Peter Koepke Tag der mündlichen Prüfung: 13. April 2012 Contents Zusammenfassung / Summary 0 Introduction 2 1 Topological Set Theories 5 1.1 Atoms, sets and classes . .5 1.2 Essential Set Theory . .6 1.3 Ordinal Numbers . 12 1.4 Pristine Sets and Inner Models . 19 1.5 Positive Specification . 25 1.6 Regularity and Union . 29 1.7 Uniformization . 31 1.8 Compactness . 37 2 Models of Topological Set Theories 39 2.1 Hyperuniverses . 39 2.2 Mild Ineffability and Topology . 42 2.3 Categories of κ-topological spaces . 48 2.4 Ultrametric Spaces and Generalized Cantor Cubes . 54 2.5 Direct Limit Models . 61 2.6 Inverse Limit Models . 69 2.7 Metric Spaces and the Hilbert Cube . 80 Open Questions 83 0 Zusammenfassung Wir stellen ein neues mengentheoretisches Axiomensystem vor, hinter welchem eine topolo- gische Intuition steht: Die Menge der Teilmengen einer Menge ist eine Topologie auf dieser Menge. Einerseits ist dieses System eine gemeinsame Abschwächung der Zermelo-Fraenkel- schen Mengenlehre ZF, der positiven Mengenlehre GPK+ und der Theorie der Hyperuni- versen; andererseits erhält es größtenteils die Ausdruckskraft dieser Theorien und hat dieselbe Konsistenzstärke wie ZF. Wir heben das zusätzliche Axiom1 einer universellen Menge als das- jenige heraus, das die Konsistenzstärke zu der von GPK+ erhöht und untersuchen weitere Axiome und Beziehungen zwischen diesen Theorien. 1 Hyperuniversen sind eine natürliche Klasse von Modellen für Theorien mit einer universellen Menge. Die @0- und @1-dimensionalen Cantorwürfel sind Beispiele von Hyperuniversen mit Additivität @0, da sie homöomorph zu ihrem Exponentialraum sind. Wir beweisen, dass im Bereich der Räume mit überabzählbarer Additivität die entsprechend verallgemeinerten Cantorwürfel diese Eigenschaft nicht haben. Zum Schluss stellen wir zwei komplementäre Konstruktionen von Hyperuniversen vor, die einige in der Literatur vorkommende Konstruktionen verallgemeinern sowie initiale und ter- minale Hyperuniversen ergeben. 1 Summary We give a new set theoretic system of axioms motivated by a topological intuition: The set of subsets of any set is a topology on that set. On the one hand, this system is a common weakening of Zermelo-Fraenkel set theory ZF, the positive set theory GPK+ and the theory of hyperuniverses. On the other hand, it retains most of the expressiveness of these theories and has the same consistency strength as ZF. We single out the additional axiom1 of the universal set as the one that increases the consistency strength to that of GPK+ and explore several other axioms and interrelations between those theories. 1 Hyperuniverses are a natural class of models for theories with a universal set. The @0- and @1-dimensional Cantor cubes are examples of hyperuniverses with additivity @0, because they are homeomorphic to their hyperspace. We prove that in the realm of spaces with uncountable additivity, none of the generalized Cantor cubes has that property. Finally, we give two complementary constructions of hyperuniverses which generalize many of the constructions found in the literature and produce initial and terminal hyperuniverses. 2 Introduction The mathematician considers collections of mathematical objects to be objects themselves. A formula φ(x) with one free variable x divides the mathematical universe into those objects to which it applies and those to which it does not. The collection of all x satisfying φ(x), the class fx j φ(x)g is routinely dealt with as if it were an object itself, a set, which can in turn be a member of yet another set. Russell’s antinomy is one of several paradoxes which show that this is not possible for all formulas φ. Specifically, the class fx j x2 = xg cannot be a set, or otherwise contradictions arise1. An axiomatic set theory can be thought of as an effort to make precise which classes are sets. It simultaneously aims at providing enough freedom of construction for all of classical mathematics and still remain consistent. It therefore must imply that all “reasonable” class comprehensions fx j φ(x)g produce sets and explain why fx j x2 = xg does not. The answer given by Zermelo-Fraenkel set theory (ZF), the most widely used and most deeply studied system of axioms for sets, is the Limitation of Size Principle: Only small classes are sets. x2 = x holds true for too many x, and no single set can comprehend all of them. On the other hand, if a is a set, every subclass fx j φ(x) ^ x2ag also is. However, the totality of all mathematical objects, the universe V = fx j x=xg, is a proper class in ZF. Since V is of considerable interest for the set theorist, several alternative axiom systems without this perceived shortcoming have been proposed. W. V. Quine’s set theory New Foundations (NF), probably the most famous one, is built around a different comprehension scheme: The existence of fx j φ(x)g is postulated only for stratified2 formulas φ(x), avoiding circularities like x 2 x but still admitting x = x. So although the Russell class is a proper class, its superclass V is a set. An interesting peculiarity of NF is that it has been shown by R. B. Jensen in [Jen68] to be consistent relative to ZF (and even much weaker theories), but only if one admits atoms3, objects which are not classes. With the condition that every object is a class, there is no known upper bound to its consistency strength. This thesis is concerned with a third family of set theories originating from yet another possi- ble answer to the paradoxes, or rather from two independent answers: Firstly, instead of the missing delimitation x2a or the circularity, one might blame the nega- tion in the formula x2 = x for Russell’s paradox. The collection of generalized positive formulas is recursively defined by several construction steps not including negation. If the existence of fx j φ(x)g is stipulated for every generalized positive formula, a beautiful “positive” set theory emerges. 1fx j x=2xg 2 fx j x=2xg if and only if fx j x=2xg 2= fx j x=2xg, by definition. 2 A formula is stratified if natural numbers l(x) can be assigned to its variables x in such a way that l(x) = l(y) for each subformula x = y, and l(x) + 1 = l(y) for every subformula x 2 y. 3also called urelements 3 Secondly, instead of demanding that every class is a set, one might settle for the ability to approximate it by a least superset, a closure in a topological sense. Surprisingly such “topological” set theories tend to prove the comprehension principle for generalized positive formulas, and conversely, in positive set theory, the universe is a topolog- ical space. More precisely, the sets are closed with respect to intersections and finite unions, and the universe is a set itself, so the sets represent the closed subclasses of a topology on V. A class is a set if and only if it is topologically closed. The first model of such a theory was constructed by R. J. Malitz in [Mal76] under the condi- tion of the existence of certain large cardinal numbers. E. Weydert, M. Forti and R. Hinnion were able to show in [Wey89, FH89] that in fact a weakly compact cardinal suffices. In [Ess97] and [Ess99], O. Esser exhaustively answered the question of consistency for a specific positive set theory, GPK+ with a choice principle, and showed that it is mutually interpretable with a variant of Kelley-Morse set theory. 1 All known models of positive set theory are hyperuniverses: κ-compact κ-topological Haus- dorff spaces homeomorphic to their own hyperspace4. These structures have been extensively studied: In [FHL96], M. Forti, F. Honsell and M. Lenisa give several equivalent definitions. Forti and Honsell discovered a much more general construction of hyperuniverses described in [FH96b], yielding among other examples structures with arbitrary given κ-compact sub- spaces. Finally, O. Esser in [Ess03] identifies the existence of mildly ineffable cardinals as equivalent to the existence of hyperuniverses with a given weight and additivity. The fact that all known models are hyperuniverses and the concern that axiom schemes given by purely syntactical requirements do not have an immediately clear intuitive meaning, mo- tivate the study of systems of topological axioms rather than theories based on any compre- hension scheme, and to axiomatize parts of the notion of a hyperuniverse “from within”. In his course “Topologische Mengenlehre” in the summer term 2006 at Ludwig-Maximilians- Universität München, H.-D. Donder gave such a topological set theory, explored its set the- oretic and topological consequences and showed that the consistency proofs of positive set theory apply to this system of axioms as well. Overview This thesis is divided into two chapters. In the first one we introduce “essential set theory” (ES), a theory in which the power set defines a topology on each set, which is not necessarily trivial. A variant of positive set theory and ZF both occur as natural extensions of ES, and in particular, essential set theory leaves open the question about a universal set, so this can be investigated separately. Also, it allows for atoms and even for the empty class ; being proper – a statement which has topologically connected models. We will show that several basic set theoretic constructions can be carried out in ES and in particular the theory of ordinal numbers is still available.
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