Fernengel, Anne: An Easton-like Theorem for all Cardinals in ZF. - Bonn, 2020. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5-58525
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5-58525,
author = {{Anne Fernengel}},
title = {An Easton-like Theorem for all Cardinals in ZF},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2020,
month = jul,

note = {We show that in Zermelo-Fraenkel 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 2κ 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 well-orderable, and there are different ways how "largeness" can now be expressed. The θ-function 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 θ-function in the theory ZF + DC + AX4. Here, the axiom AX4 is the assertion that for every cardinal λ, the set of all countable subsets of λ can be well-ordered. Together with the Axiom of Dependent Choice (DC), the theory ZF + DC + AX4 provides a rich framework for combinatorial set theory in the ¬ AC-context, 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 θ-function on a set of uncountable cardinals, we construct a model N of ZF + DC + AX4 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 θ-values of many cardinals simultaneously. Our second theorem deals with the question whether also any "reasonable" behavior of the θ-function on a class of infinite cardinals can be realized in ZF. (The construction mentioned above can not be straightforwardly generalized to a class-sized forcing notion and is therefore only suitable for treating set many θ-values 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 ZF-model N such that NV is a cardinal-preserving extension with θN(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 θ-function 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.},

url = {https://hdl.handle.net/20.500.11811/8431}

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