Iterating Symmetric Extensions

Abstract

The notion of a symmetric extension extends the usual notion of forcing by identifying a particular class of names which forms an intermediate model of $ZF$ between the ground model and the generic extension, and often the axiom of choice fails in these models. Symmetric extensions are generally used to prove choiceless consistency results. We develop a framework for iterating symmetric extensions in order to construct new models of $ZF$. We show how to obtain some well-known and lesser-known results using this framework. Specifically, we discuss Kinna–Wagner principles and obtain some results related to their failure.

How to have more things by forgetting how to count them

“…This is true since M = V(A), and the argument for this equality is the same even when using Add(ω, κ), which is easy to see from analysing the same proofs as in the case κ = ω. 5 Remark 4.4. The corollary means that the process works in reverse as well, namely, starting with Add(ω, λ) with finitary permutations of λ and a filter of subgroups generated by pointwise stabilisers of finite subsets of λ, we end up with an analogue of the Cohen model where we have a set of Cohen reals which is Dedekind-finite.…”

Realizing Realizability Results With Classical Constructions

“…Other than clarifying the train of thoughts of the author, we would like to reinforce the view that "if you never fall, you will never learn to get up" which is something many young researchers might struggle with 7. This reinforces the idea that somehow an ultrapower is involved, although this is just a place to start intuitively thinking about realizability models.…”

Critical cardinals

“…The notion of symmetrically generic filters is the one needed to interpret correctly the names in HS. This is reflected in the following theorem [4,Theorem 8.4].…”

“…Finally, while we do not discuss iterations of symmetric extensions in full, or even in the case of a two-step iteration, it will be conceptually relevant to the Silver-like criterion for lifting elementary embeddings to symmetric extensions, so we urge the reader to glance through the second author's [4]. One definition from that context is relevant to this work, and that is the generic semi-direct product of groups.…”

“…For the rest of the discussion on supercompact Radin forcing, κ will be a fixed supercompact cardinal. 4 Fix λ > κ, and let Λ : κ → κ be a function that some (2 λ <κ ) + -supercompact embedding j exists with crit(j) = κ and j(Λ)(κ) = λ. By supercompactness of κ, there is a coherent sequence of measures [5,Section 2]).…”