Formalizing Ordinal Partition Relations Using Isabelle/HOL

Abstract

This is an overview of a formalization project in the proof assistant Isabelle/HOL of a number of research results in infinitary combinatorics and set theory (more specifically in ordinal partition relations) by Erd ős-Milner, Specker, Larson and Nash-Williams, leading to Larson's proof of the unpublished result by E.C. Milner asserting that for all m ∈ N, ω ω −→ (ω ω , m). This material has been recently formalised by Paulson and is available on the Archive of Formal Proofs; here we discuss some of the most challenging aspects of the formalization process. This project is also a demonstration of working with Zermelo-Fraenkel set theory in higher-order logic.

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

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“…Then I decided to apply it to a project that had been proposed by Mirna Džamonja and Angeliki Koutsoukou-Argyraki: to formalise some ordinal partition theory. This concerns advanced generalisations of Ramsey's theorem; the formalisation [17] was complete by August 2020 and a paper is now available [4]. However, this work essentially belongs to straight set theory, borrowing little from Isabelle/HOL apart from natural numbers and lists.…”

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