Reverse Mathematics and parameter-free Transfer

TL;DR

This paper investigates the parameter-free Transfer axiom in Nonstandard Analysis, establishing a base theory for Reverse Mathematics and demonstrating that many equivalences fail without parameters.

Contribution

It introduces a base theory for parameter-free Transfer in Reverse Mathematics and analyzes the impact of parameter-free restrictions on classical equivalences.

Findings

Most classical reversals do not hold without parameters

A base theory for parameter-free Transfer is formulated

Natural reversals are proved within the new framework

Abstract

Recently, conservative extensions of Peano and Heyting arithmetic in the spirit of Nelson's axiomatic approach to Nonstandard Analysis, have been proposed. In this paper, we study the Transfer axiom of Nonstandard Analysis restricted to formulas without parameters. Based on this axiom, we formulate a base theory for the Reverse Mathematics of Nonstandard Analysis and prove some natural reversals, and show that most of these equivalences do not hold in the absence of parameter-free Transfer.