# Realizing realizability results with classical constructions

## TL;DR

This paper explores how classical set-theoretic constructions, specifically symmetric extensions, can replicate Krivine's realizability results, and introduces a new condition for preserving certain forms of choice.

## Contribution

It demonstrates the realization of Krivine's realizability results within classical symmetric extensions and proposes a novel condition for preserving specific choice principles.

## Key findings

- Successfully reproduces Krivine's realizability results in symmetric extensions
- Introduces a new condition for preserving well-ordered and other choice principles
- Provides insights into the interplay between realizability and classical set-theoretic models

## Abstract

J.L. Krivine developed a new method based on realizability to construct models of set theory where the axiom of choice fails. We attempt to recreate his results in classical settings, i.e. symmetric extensions. We also provide a new condition for preserving well-ordered, and other particular type of choice, in the general settings of symmetric extensions.