[Paper Review] Formalising Ordinal Partition Relations Using Isabelle/HOL

Mirna Džamonja, Angeliki Koutsoukou-Argyraki+1 more|arXiv (Cornell University)|Nov 26, 2020

Logic, Reasoning, and Knowledge34 references4 citations

TL;DR

This paper presents a formalization in Isabelle/HOL of key results in infinitary combinatorics, specifically ordinal partition relations, including Larson's proof of the unpublished result that $\omega^{\omega} \longrightarrow (\omega^{\omega}, m)$ for all $m \in \mathbb{N}$. The work establishes a rigorous foundation for structural Ramsey theory using higher-order logic and demonstrates the feasibility of formalizing complex set-theoretic constructions involving transfinite ordinals.

ABSTRACT

This is an overview of a formalisation 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in ordinal partition relations) by Erdős--Milner, Specker, Larson and Nash-Williams, leading to Larson's proof of the unpublished result by E.C. Milner asserting that for all $m \in \mathbb{N}$, $ω^ω\arrows(ω^ω, m)$. 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 formalisation process. This project is also a demonstration of working with Zermelo-Fraenkel set theory in higher-order logic.

Motivation & Objective

• To formalize foundational results in ordinal partition relations, a largely unexplored area in infinitary combinatorics.
• To demonstrate the viability of using higher-order logic and Isabelle/HOL for formalizing advanced set-theoretic constructions involving transfinite ordinals.
• To provide a rigorous, machine-checked proof of Larson’s result on $\omega^\omega \longrightarrow (\omega^\omega, m)$, a key result in structural Ramsey theory.
• To establish a framework for future discovery of new ordinal partition relations using formal methods.
• To address the lack of intuition and high error rate in this field by providing a verified formal foundation.

Proposed method

• Employing the Isabelle/HOL proof assistant to formalize definitions and proofs from set theory and infinitary combinatorics.
• Defining a novel interaction scheme $i_k(U)$ that uniquely maps sets of lists to interaction sequences, ensuring injectivity via structural induction.
• Using inductive definitions to formalize the concept of thin sets and initial segments in list structures.
• Proving the uniqueness of the interaction scheme through a 240-line formal proof, relying on structural properties of list concatenation and ordering.
• Encoding ordinal arithmetic and partition relations using higher-order logic, with $\omega$ identified as the set of natural numbers.
• Verifying key lemmas such as the thinness of interaction scheme sets and the injectivity of the scheme under specific structural constraints.

Experimental results

Research questions

• RQ1 Can complex results in ordinal partition relations, such as $\omega^\omega \longrightarrow (\omega^\omega, m)$, be formally verified using interactive theorem proving?
• RQ2 How can Zermelo-Fraenkel set theory be effectively formalized within higher-order logic for transfinite constructions?
• RQ3 What structural properties of list-based interaction schemes ensure their injectivity and uniqueness in the context of ordinal partition relations?
• RQ4 To what extent can formalization tools like Isabelle/HOL support the discovery of new partition relations in structural Ramsey theory?
• RQ5 Why do standard set-theoretic methods fail for ordinal partition relations, and can formalization help overcome this?

Key findings

• The interaction scheme $i_k(U)$ is formally proven to be uniquely defined for all valid inputs, with a 240-line proof establishing the uniqueness of the interaction sequence.
• The set of all interaction schemes $\{i_l(U) \mid U \text{ has form } l\}$ is formally shown to be thin, a critical technical lemma in Larson’s proof.
• The formalization confirms that $\omega^\omega \longrightarrow (\omega^\omega, m)$ holds for all $m \in \mathbb{N}$, verifying a previously unpublished result by E.C. Milner.
• The proof of injectivity of the interaction scheme is established via a 50-line inductive argument on list lengths, relying on strict sorting and concatenation properties.
• The formalization demonstrates that standard set-theoretic tools like countable elementary submodels are ineffective for reasoning about countable ordinals, justifying the need for direct formalization.
• The work provides a verified foundation for future research in structural Ramsey theory, particularly for discovering new ordinal partition relations through computational methods.