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

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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 ZermeloFraenkel 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_1dimensional 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:  Thesis (Dissertation, LMU Munich) 

Keywords:  essential set theory,hyperuniverse,set theory,topological set theory,universal set 
Subjects:  600 Natural sciences and mathematics > 510 Mathematics 600 Natural sciences and mathematics 
Faculties:  Faculty of Mathematics, Computer Science and Statistics 
Language:  English 
Date Accepted:  13. April 2012 
1. Referee:  Donder, HansDieter 
Persistent Identifier (URN):  urn:nbn:de:bvb:19142582 
MD5 Checksum of the PDFfile:  92d3089afef25fe32552e776341f1e86 
Signature of the printed copy:  0001/UMC 20300 
ID Code:  14258 
Deposited On:  11. May 2012 09:18 
Last Modified:  16. Oct 2012 09:00 