Abstraction Principles and the Classification of Second-Order Equivalence Relations

TL;DR

This paper enhances classification theorems for second-order equivalence relations, enabling better analysis of models in abstraction principles and advancing understanding in neo-logicist philosophy of mathematics.

Contribution

It improves Fine's classification theorem for equivalence relations in second-order logic and extends Walsh and Ebels-Duggan's categoricity results.

Findings

Equivalence classes have only three possible bicardinal profiles.

Improved theorems facilitate analysis of models in abstraction principles.

Enhanced categoricity results for well-behaved models.

Abstract

This paper improves two existing theorems of interest to neo-logicist philosophers of mathematics. The first is a classification theorem due to Fine for equivalence relations between concepts definable in a well-behaved second-order logic. The improved theorem states that if an equivalence relation $E$ is defined without non-logical vocabulary, then the bicardinal slice of any equivalence class---those equinumerous elements of the equivalence class with equinumerous complements---can have one of only three profiles. The improvements to Fine's theorem allow for an analysis of the well-behaved models had by an abstraction principle, and this in turn leads to an improvement of Walsh and Ebels-Duggan's relative categoricity theorem.