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.

35 pages+1 page appendix

References in corpus (1)

• The Bristol Model: an abyss called a Cohen real

Cited by in corpus (11)

• Borel reducibility and symmetric models
• Combinatorial Properties and Dependent choice in symmetric extensions based on Lévy Collapse
• Critical Cardinals
• How to have more things by forgetting how to count them
• Realizing realizability results with classical constructions
• Choiceless Chain Conditions
• Canonical Truth
• Does $\mathsf{DC}$ imply ${\mathsf{AC}}_{\omega }$, uniformly?
• Small measurable cardinals
• The Morris model
• Intermediate models and Kinna--Wagner Principles