The formal verification of the ctm approach to forcing

TL;DR

This paper presents a computer-verified proof using Isabelle/ZF for constructing generic extensions of set models that satisfy both CH and its negation, advancing formal methods in set theory and forcing.

Contribution

It introduces a formal verification of the ctm approach to forcing, including a specific subset and function to control extensions satisfying CH and its negation.

Findings

Successfully verified the construction of generic extensions with and without CH

Developed a formal framework isolating key axioms for forcing extensions

Implemented the proof in Isabelle/ZF, demonstrating the feasibility of formal set-theoretic proofs

Abstract

We discuss some highlights of our computer-verified proof of the construction, given a countable transitive set-model $M$ of $\mathit{ZFC}$, of generic extensions satisfying $\mathit{ZFC}+\neg\mathit{CH}$ and $\mathit{ZFC}+\mathit{CH}$. Moreover, let $\mathcal{R}$ be the set of instances of the Axiom of Replacement. We isolated a 21-element subset $\Omega\subseteq\mathcal{R}$ and defined $\mathcal{F}:\mathcal{R}\to\mathcal{R}$ such that for every $\Phi\subseteq\mathcal{R}$ and $M$-generic $G$, $M\models \mathit{ZC} \cup \mathcal{F}\text{``}\Phi \cup \Omega$ implies $M[G]\models \mathit{ZC} \cup \Phi \cup \{ \neg \mathit{CH} \}$, where $\mathit{ZC}$ is Zermelo set theory with Choice. To achieve this, we worked in the proof assistant Isabelle, basing our development on the Isabelle/ZF library by L. Paulson and others.