Comparing Hume's Principle, Basic Law V and Peano Arithmetic

TL;DR

This paper compares the interpretability strength of models of Hume's Principle and Basic Law V using hyperarithmetic theory, revealing that only certain fragments can interpret second-order Peano arithmetic.

Contribution

It introduces new model constructions with restricted comprehension and analyzes their interpretability bounds within subsystems of second-order arithmetic.

Findings

A consistent extension of Basic Law V interprets hyperarithmetic second-order Peano arithmetic.

Hume's Principle's hyperarithmetic fragment does not interpret second-order Peano arithmetic.

The results establish bounds on the interpretability strength of these theories.

Abstract

This paper presents new constructions of models of Hume's Principle and Basic Law V with restricted amounts of comprehension. The techniques used in these constructions are drawn from hyperarithmetic theory and the model theory of fields, and formalizing these techniques within various subsystems of second-order Peano arithmetic allows one to put upper and lower bounds on the interpretability strength of these theories and hence to compare these theories to the canonical subsystems of second-order arithmetic. The main results of this paper are: (i) there is a consistent extension of the hyperarithmetic fragment of Basic Law V which interprets the hyperarithmetic fragment of second-order Peano arithmetic, and (ii) the hyperarithmetic fragment of Hume's Principle does not interpret the hyperarithmetic fragment of second-order Peano arithmetic, so that in this specific sense there is no predicative version of Frege's Theorem.