Sort logic and foundations of mathematics

TL;DR

This paper advocates for sort logic as a robust foundation for mathematics, replacing ad hoc assumptions with ZFC-like axioms while maintaining a structuralist perspective and offering strong model-theoretic capabilities.

Contribution

It introduces sort logic as an extension of second order logic that avoids ad hoc large domain assumptions by using set theory-like axioms, strengthening the foundation of mathematics.

Findings

Sort logic is the strongest model-theoretic logic.

Every set-theoretic model class is definable in sort logic.

Sort logic enables formulation of strong reflection principles.

Abstract

I have argued elsewhere that second order logic provides a foundation for mathematics much in the same way as set theory does, despite the fact that the former is second order and the latter first order, but second order logic is marred by reliance on ad hoc {\em large domain assumptions}. In this paper I argue that sort logic, a powerful extension of second order logic, provides a foundation for mathematics without any ad hoc large domain assumptions. The large domain assumptions are replaced by ZFC-like axioms. Despite this resemblance to set theory sort logic retains the structuralist approach to mathematics characteristic of second order logic. As a model-theoretic logic sort logic is the strongest logic. In fact, every model class definable in set theory is the class of models of a sentence of sort logic. Because of its strength sort logic can be used to formulate particularly strong reflection principles in set theory.