Fernengel, Anne: An Easton-like Theorem for all Cardinals in ZF. - Bonn, 2020. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.

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@phdthesis{handle:20.500.11811/8431,

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 (

Without the Axiom of Choice (

Our first theorem answers a question of Saharon Shelah, who asked whether there are any bounds on the θ-function in the theory

Our forcing notion is based on ideas from the paper

Given a ground model

We introduce a new notion of class forcing ℙ, consisting of functions on trees with finitely many maximal points. Our eventual model

We conclude that indeed, any "reasonable" behavior of the θ-function can be realized in

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

}

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 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 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 + AX**. Here, the axiom_{4}**AX**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 (_{4}**DC**), the theory**ZF + DC + AX**provides a rich framework for combinatorial set theory in the_{4}**¬ 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 + AX**where this behavior is realized._{4}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**N**⊇**V**is a cardinal-preserving 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 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}

}