Formalization of Forcing in Isabelle/ZF

Abstract

We formalize the theory of forcing in the set theory framework of Isabelle/ZF. Under the assumption of the existence of a countable transitive model of ZFC , we construct a proper generic extension and show that the latter also satisfies ZFC . In doing so, we remodularized Paulson's ZF-Constructibility library.

Wetzel: Formalisation of an Undecidable Problem Linked to the Continuum Hypothesis

“…During the 1990s, I made some significant formal developments using Isabelle/ZF [14], culminating in a proof of the relative consistency of the axiom of choice using Gödel's constructible universe [15]. Recently, several highly impressive formalisations of forcing were done within Isabelle/ZF [8,9].…”

A Formalised Theorem in the Partition Calculus

“…Isabelle is an interactive theorem prover based on a logical framework: a minimal formalism intended for representing formal proofs in a variety of logics [28]. Isabelle/ZF supports first-order logic and set theory, and has been used to formalise the constructible universe [22] and forcing [29]. But its most popular instance by far is Isabelle/HOL [11], supporting higher-order logic.…”