Set Theory for Verification: I. From Foundations to Functions

TL;DR

This paper develops a set-theoretic logic for verification derived from Zermelo-Fraenkel axioms, utilizing Isabelle proof assistant to demonstrate its flexibility and effectiveness in formal proofs of classical theorems.

Contribution

It introduces a set-theoretic logic framework within Isabelle, enabling formal verification with support for various set theory variants and complex proof strategies.

Findings

Successfully derived rules for descriptions, relations, and functions

Interactive proofs of Cantor's Theorem and Ramsey's Theorem

Demonstrated Isabelle's adaptability to different set theory variants

Abstract

A logic for specification and verification is derived from the axioms of Zermelo-Fraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higher-order syntax supports the definition of new binding operators. Unknowns in subgoals can be instantiated incrementally. The paper describes the derivation of rules for descriptions, relations and functions, and discusses interactive proofs of Cantor's Theorem, the Composition of Homomorphisms challenge [9], and Ramsey's Theorem [5]. A generic proof assistant can stand up against provers dedicated to particular logics.