An Easton-like Theorem for all Cardinals in ZF

Abstract

We show that in Zermelo-Fraenkel Set Theory without the Axiom of Choice (<strong>ZF</strong>), a surjectively modified Continuum Function θ(κ) can take almost arbitrary values on all cardinals. This is in sharp contrast to the situation in <strong>ZFC</strong>, where on the one hand, Easton's Theorem states that the Continuum Function on the class of all <em>regular</em> cardinals is essentially undetermined, but on the other hand, various results show that the value of 2^κ for <em>singular</em> cardinals κ is strongly influenced by the behavior of the Continuum Function below. <br /> Without the Axiom of Choice (<strong>AC</strong>), 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 <em>surjective size</em> of the powersets. <br /> Our first theorem answers a question of Saharon Shelah, who asked whether there are any bounds on the θ-function in the theory <strong>ZF + DC + AX₄</strong>. Here, the axiom <strong>AX₄</strong> 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 (<strong>DC</strong>), the theory <strong>ZF + DC + AX₄ </strong> provides a rich framework for combinatorial set theory in the <strong>¬ AC</strong>-context, in which set theory is "not so far from normal" (Shelah). Nevertheless, we prove that the answer to Shelah's question is <em>no</em>: Given any "reasonable" behavior of the θ-function on a set of uncountable cardinals, we construct a model <strong>N</strong> of <strong>ZF + DC + AX₄ </strong> where this behavior is realized. <br /> Our forcing notion is based on ideas from the paper <em>"Violating the Singular Cardinals Hypothesis without Large Cardinals"</em> (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 <em>class</em> of infinite cardinals can be realized in <strong>ZF</strong>. (The construction mentioned above can not be straightforwardly generalized to a class-sized forcing notion and is therefore only suitable for treating <em>set</em> many θ-values at the same time.) <br /> Given a ground model <strong>V</strong> 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 <strong>ZF</strong>-model <strong>N</strong> such that <strong>N</strong>⊇<strong>V</strong> is a cardinal-preserving extension with θ^{<strong>N</strong>}(κ) = F(κ) for all cardinals κ? <br /> We introduce a new notion of class forcing ℙ, consisting of functions on trees with finitely many maximal points. Our eventual model <strong>N</strong> is a symmetric extension by this class forcing ℙ. <br /> We conclude that indeed, any "reasonable" behavior of the θ-function can be realized in <strong>ZF</strong> -- the only restrictions are the obvious ones. In other words: An analogue of Easton's theorem holds for all cardinals.