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Upwards homogeneity in iterated symmetric extensions

Calliope Ryan-Smith, Jonathan Schilhan, Yujun Wei

TL;DR

The paper identifies upwards homogeneity as a precise criterion for iterated symmetric extensions that prevents the second stage from adding new subsets of the ground model, formalizing a condition under which $\mathcal{P}(V)^M=\mathcal{P}(V)^N$ holds in intermediate models. It develops the framework for two-step and more general iterations, illustrating the method with canonical constructions such as Monro's iteration, Morris's iteration, and the Bristol model, and it shows how upwards homogeneity can be preserved through complex limit-stage iterations. The work clarifies when forcing over symmetric extensions preserves ground-model structure, introduces mechanisms like predense witness collections and permutable scales, and provides open questions about extending UPH to broader contexts and higher rank levels. Overall, the results deepen the understanding of how symmetry and iteration interact to constrain the propagation of ground-model subsets in choiceless set-theoretic constructions, with potential implications for the study of Kinna–Wagner principles and related independence phenomena.

Abstract

It is sometimes desirable in choiceless constructions of set theory that one iteratively extends some ground model without adding new sets of ordinals after the first extension. Pushing this further, one may wish to have models $V \subseteq M \subseteq N$ of $\mathsf{ZF}$ such that $N$ contains no subsets of $V$ that do not already appear in $M$. We isolate, in the case that $M$ and $N$ are symmetric extensions (particular inner models of a generic extension of $V$), the exact conditions that cause this behaviour and show how it can broadly be applied to many known constructions. We call this behaviour upwards homogeneity.

Upwards homogeneity in iterated symmetric extensions

TL;DR

The paper identifies upwards homogeneity as a precise criterion for iterated symmetric extensions that prevents the second stage from adding new subsets of the ground model, formalizing a condition under which holds in intermediate models. It develops the framework for two-step and more general iterations, illustrating the method with canonical constructions such as Monro's iteration, Morris's iteration, and the Bristol model, and it shows how upwards homogeneity can be preserved through complex limit-stage iterations. The work clarifies when forcing over symmetric extensions preserves ground-model structure, introduces mechanisms like predense witness collections and permutable scales, and provides open questions about extending UPH to broader contexts and higher rank levels. Overall, the results deepen the understanding of how symmetry and iteration interact to constrain the propagation of ground-model subsets in choiceless set-theoretic constructions, with potential implications for the study of Kinna–Wagner principles and related independence phenomena.

Abstract

It is sometimes desirable in choiceless constructions of set theory that one iteratively extends some ground model without adding new sets of ordinals after the first extension. Pushing this further, one may wish to have models of such that contains no subsets of that do not already appear in . We isolate, in the case that and are symmetric extensions (particular inner models of a generic extension of ), the exact conditions that cause this behaviour and show how it can broadly be applied to many known constructions. We call this behaviour upwards homogeneity.
Paper Structure (17 sections, 6 theorems, 24 equations, 1 figure)

This paper contains 17 sections, 6 theorems, 24 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Preliminaries
  4. Symmetric extensions
  5. Iterating symmetric extensions
  6. Wreath products
  7. Upwards homogeneity
  8. Examples
  9. Monro's iteration
  10. Forcing over symmetric extensions
  11. Morris's iteration
  12. The Bristol model
  13. Iterating the systems $\dot\mathscr{T}_\gamma$
  14. Constructing $\mathscr{T}_\gamma$
  15. Upwards homogeneity of $\mathscr{S}_\alpha\ast\dot{\mathscr{T}}_\alpha$
  16. ...and 2 more sections

Key Result

Theorem 2.1

Let $\pi\in \mathop{\mathrm{Aut}}\nolimits(\mathbb{P})$. Then for any $\mathbb{P}$-name $\dot{x}$ and formula $\varphi$, $p\mathrel{\Vdash} \varphi(\dot{x})$ if and only if $\pi p\mathrel{\Vdash}\varphi(\pi\dot{x})$.∎

Figures (1)

  • Figure 1: The action of a wreath product: The co-ordinate $\langle x,y\rangle$ is transformed by $\pi=(g^\ast,(h_x^\ast)_{x\in X})$ first according to $h_x^\ast$ into $\langle x,h_x^\ast(y)\rangle$, and then according to $g^\ast$ into $\langle g^\ast x,h_x^\ast(y)\rangle$.

Theorems & Definitions (26)

  • Definition
  • Remark
  • Theorem 2.1: Symmetry Lemma
  • Example 2.2
  • Claim 2.2.1
  • proof : Proof of Claim
  • Definition 2.3: Cohen forcing
  • Definition 2.4: Two-step symmetric extension
  • Definition 2.5
  • Definition 3.1
  • ...and 16 more