The Strength of Abstraction with Predicative Comprehension

TL;DR

This paper investigates the strength of abstraction principles under predicative restrictions, demonstrating that a certain predicative Fregean theory can interpret second-order Peano arithmetic, thus revealing the power of abstraction in a predicative setting.

Contribution

It introduces a predicative Fregean theory containing all abstraction principles with provably equivalence relations, and shows it can interpret second-order Peano arithmetic.

Findings

Predicative Fregean theory interprets second-order Peano arithmetic.

Abstraction principles with provable equivalence relations are sufficient in a predicative context.

The study clarifies the role of abstraction in foundational mathematics.

Abstract

Frege's theorem says that second-order Peano arithmetic is interpretable in Hume's Principle and full impredicative comprehension. Hume's Principle is one example of an abstraction principle, while another paradigmatic example is Basic Law V from Frege's Grundgesetze. In this paper we study the strength of abstraction principles in the presence of predicative restrictions on the comprehension schema, and in particular we study a predicative Fregean theory which contains all the abstraction principles whose underlying equivalence relations can be proven to be equivalence relations in a weak background second-order logic. We show that this predicative Fregean theory interprets second-order Peano arithmetic.