The technique of symmetric extensions is derived from forcing and it is one of the most important tools for studying models without the Axiom of Choice. Despite being incredibly successful since the 1960s, our understanding of the technique remained fairly limited compared to the theory of forcing. Whereas forcing developed products and iterations, no serious attempts at developing any general framework for iterating symmetric extensions were presented before [10], where only finite support iterations are treated. In this paper we develop the theory of symmetric extensions including different types of iterations, quotients, equivalents, and the structural results that can be described in this language. In particular, we give a modern exposition to some of the important theorems of Grigorieff [3], study Kinna--Wagner Principles in symmetric extensions, and show that it is provable from $\mathsf{ZF}$ that every set lies in a symmetric extension of $\operatorname{HOD}$.
Iteration is a common methodology in mathematics. It allows us to solve a large problem by breaking it into many small problems, and then solve one problem at a time. Set theory is not different, and shortly after Cohen published his two papers on forcing [1,2] the first instances of iterated forcing were discovered as well. Most famously, Solovay and Tennenbaum [18] proved the consistency of the Suslin Hypothesis using iterated forcing, and the technique had been extended significantly in the decades since its conception.
Forcing is very useful for the study of models of ZFC, as it preserves the Axiom of Choice, 1 which is certainly a desirable feature, but it makes studying choiceless universes more difficult. Cohen, however, already noticed in his initial work that one can pass to an inner model of the extension, an intermediate model, where the Axiom of Choice fails. This eventually gave rise to the technique of symmetric extensions where in addition to the forcing notion we are given an automorphism group and a filter of subgroups, which then allow us to identify in advance an intermediate model of ZF and use forcing-theoretic tools to study it. This concept was studied more abstractly by Grigorieff [3], where several structural results were obtained about symmetric extensions.
Iterating symmetric extensions was something that was present in the mathematical literature for decades, but until [10] this was always in ad-hoc form and never in a generalized framework. Incredible examples include the works in [14, 16, 17].
A first attempt to design a general framework for iterating symmetric extension was published in [10] by the first author of this paper. The presentation of the framework is certainly difficult, as its use is still limited. Moreover, the framework only allows finite support iterations which makes it harder to preserve certain choice principles, such as Dependent Choice which naturally lends itself to be preserved by countably closed extensions.
In this paper we extend and rework the basic theory of symmetric extensions and the iteration framework. Indeed, we aim to streamline the definitions and notation from [10] to make the application of the method much simpler. The goal of this manuscript is to make symmetric extensions and their extensions more accessible.
We develop the theory of symmetric extensions in parallel to the theory of forcing. We study the basics of symmetric extensions, iterations and products, as well as an intermediate notion of reduced products. 2 These lead naturally to completions, embeddings, and quotients of symmetric systems.
All these ideals provide us with a new and more accessible language to use and present refreshed and generalized theorems from Grigorieff’s original paper. In turn, this allows us to prove that starting from a model of ZFC, every symmetric extension has an associated cardinal κ, such that KWP * κ + must hold in the symmetric extension (see Theorem 11.7). We also this new language to prove that every set lies in a symmetric extension of HOD (see Theorem 12.2).
In this section we will reintroduce symmetric systems and symmetric extensions using new notation and terminology which we think is more streamlined and we expect will be more conducive for future work. We follow the standard treatment of forcing, wherein a notion of forcing is a preordered set P with a maximum element, often denoted by 1 P . The elements of P are called conditions and we write q ≤ p to mean that q is a stronger condition than p, or that q extends p. Two conditions are compatible if they have a common extension, and otherwise they are incompatible.
The class of P-names, V P is defined recursively by V P α = {P(P × V P β ) : β < α}, with V P = α∈Ord V P α . Given a set of P-names, X, we write X • to denote the “obvious” name it generates. Namely,
We extend this notation to ordered pairs, as well as sequences and functions whose domains are •-names. Using this notation we can define the canonical ground model names, x = {y : y ∈ x} • .
2.1. Symmetric systems. Definition 2.1. Let G be a group. Then a set F of subgroups of G is a filter if
F is normal if for every π ∈ G, H ∈ F, also πHπ -1 ∈ F. Definition 2.2. A symmetric system is a triple S = (P, G, F ) where P is a forcing notion, G is a group of automorphisms on P and F is a normal filter of subgroups of G.
In many situations, we may in fact allow G not to literally consist of automorphisms on P. Rather sometimes there will merely be an implied action of G on P. This is a purely technical distinction that does not affect any of the arguments or the practice of symmetric extensions. In particular, all the notions the we define below can be derived in an analogous way for this more general situation. The official definition remains Definition 2.2.
Whenever π is an automorphism of P, π naturally extends to P-names by a recursive definition: π( ẋ) = {(π(p), π( ẏ)) : (p, ẏ) ∈ ẋ}.
This action is well-behaved wit
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