An Eastonlike Theorem for all Cardinals in ZF
An Eastonlike Theorem for all Cardinals in ZF
dc.contributor.advisor  Koepke, Peter  
dc.contributor.author  Fernengel, Anne  
dc.date.accessioned  20200701T09:50:53Z  
dc.date.available  20200701T09:50:53Z  
dc.date.issued  01.07.2020  
dc.identifier.uri  https://hdl.handle.net/20.500.11811/8431  
dc.description.abstract  We show that in ZermeloFraenkel Set Theory without the Axiom of Choice (ZF), a surjectively modified Continuum Function θ(κ) can take almost arbitrary values on all cardinals. This is in sharp contrast to the situation in ZFC, where on the one hand, Easton's Theorem states that the Continuum Function on the class of all regular cardinals is essentially undetermined, but on the other hand, various results show that the value of 2^{κ} for singular cardinals κ is strongly influenced by the behavior of the Continuum Function below.
Without the Axiom of Choice (AC), the powerset of a cardinal is generally not wellorderable, and there are different ways how "largeness" can now be expressed. The θfunction maps any cardinal κ to the least cardinal α for which there is no surjective function from the powerset of κ onto α, thus measuring the surjective size of the powersets. Our first theorem answers a question of Saharon Shelah, who asked whether there are any bounds on the θfunction in the theory ZF + DC + AX_{4}. Here, the axiom AX_{4} is the assertion that for every cardinal λ, the set of all countable subsets of λ can be wellordered. Together with the Axiom of Dependent Choice (DC), the theory ZF + DC + AX_{4} provides a rich framework for combinatorial set theory in the ¬ ACcontext, in which set theory is "not so far from normal" (Shelah). Nevertheless, we prove that the answer to Shelah's question is no: Given any "reasonable" behavior of the θfunction on a set of uncountable cardinals, we construct a model N of ZF + DC + AX_{4} where this behavior is realized. Our forcing notion is based on ideas from the paper "Violating the Singular Cardinals Hypothesis without Large Cardinals" (2012) by Moti Gitik and Peter Koepke. We modify and generalize their construction in order to treat the θvalues of many cardinals simultaneously. Our second theorem deals with the question whether also any "reasonable" behavior of the θfunction on a class of infinite cardinals can be realized in ZF. (The construction mentioned above can not be straightforwardly generalized to a classsized forcing notion and is therefore only suitable for treating set many θvalues at the same time.) Given a ground model V with a function F on the class of infinite cardinals such that F is weakly monotone and F(κ) ≥ κ^{++} holds for all κ, is there a ZFmodel N such that N⊇V is a cardinalpreserving extension with θ^{N}(κ) = F(κ) for all cardinals κ? We introduce a new notion of class forcing ℙ, consisting of functions on trees with finitely many maximal points. Our eventual model N is a symmetric extension by this class forcing ℙ. We conclude that indeed, any "reasonable" behavior of the θfunction can be realized in ZF  the only restrictions are the obvious ones. In other words: An analogue of Easton's theorem holds for all cardinals.  en 
dc.language.iso  eng  
dc.rights  In Copyright  
dc.rights.uri  http://rightsstatements.org/vocab/InC/1.0/  
dc.subject  Kontinuumsfunktion  
dc.subject  singuläreKardinalzahlenHypothese (SCH)  
dc.subject  Prinzip der abhängigen Wahlen (DC)  
dc.subject  ThetaFunktion  
dc.subject  Symmetrisches Modell  
dc.subject  Satz von Easton  
dc.subject  Continuum Function  
dc.subject  Singular Cardinals Hypothesis (SCH)  
dc.subject  The Axiom of Dependent Choice (DC)  
dc.subject  Theta Function  
dc.subject  Symmetric Model  
dc.subject  Easton's Theorem  
dc.subject.ddc  510 Mathematik  
dc.title  An Eastonlike Theorem for all Cardinals in ZF  
dc.type  Dissertation oder Habilitation  
dc.publisher.name  Universitäts und Landesbibliothek Bonn  
dc.publisher.location  Bonn  
dc.rights.accessRights  openAccess  
dc.identifier.urn  https://nbnresolving.org/urn:nbn:de:hbz:558525  
ulbbn.pubtype  Erstveröffentlichung  
ulbbnediss.affiliation.name  Rheinische FriedrichWilhelmsUniversität Bonn  
ulbbnediss.affiliation.location  Bonn  
ulbbnediss.thesis.level  Dissertation  
ulbbnediss.dissID  5852  
ulbbnediss.date.accepted  28.01.2020  
ulbbnediss.institute  MathematischNaturwissenschaftliche Fakultät : Fachgruppe Mathematik / Mathematisches Institut  
ulbbnediss.fakultaet  MathematischNaturwissenschaftliche Fakultät  
dc.contributor.coReferee  Lücke, Philipp 
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