The strength of replacement in weak arithmetic

TL;DR

This paper investigates the independence of the replacement axiom scheme from weak theories of arithmetic, demonstrating its independence under certain complexity assumptions.

Contribution

It establishes the independence of the replacement scheme from various weak arithmetic theories, sometimes relying on complexity assumptions.

Findings

Replacement scheme is independent of some weak theories.

Independence results depend on complexity assumptions.

Provides new insights into the structure of weak arithmetic theories.

Abstract

The replacement (or collection or choice) axiom scheme asserts bounded quantifier exchange. We prove the independence of this scheme from various weak theories of arithmetic, sometimes under a complexity assumption.