Fackler, Andreas (2012): Topological set theories and hyperuniverses. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics |

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**DOI**: 10.5282/edoc.14258

### Abstract

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 axiom 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. Hyperuniverses are a natural class of models for theories with a universal set. The Aleph_0- and Aleph_1-dimensional Cantor cubes are examples of hyperuniverses with additivity Aleph_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.

Item Type: | Theses (Dissertation, LMU Munich) |
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Keywords: | essential set theory,hyperuniverse,set theory,topological set theory,universal set |

Subjects: | 500 Natural sciences and mathematics > 510 Mathematics 500 Natural sciences and mathematics |

Faculties: | Faculty of Mathematics, Computer Science and Statistics |

Language: | English |

Date of oral examination: | 13. April 2012 |

1. Referee: | Donder, Hans-Dieter |

MD5 Checksum of the PDF-file: | 92d3089afef25fe32552e776341f1e86 |

Signature of the printed copy: | 0001/UMC 20300 |

ID Code: | 14258 |

Deposited On: | 11. May 2012 09:18 |

Last Modified: | 24. Oct 2020 02:48 |