Nonstandard Analysis and Constructivism!

TL;DR

This paper examines the relationship between Nonstandard Analysis and constructivism, showing how some theorems can be made constructive while refuting claims about certain axioms' constructive validity.

Contribution

It clarifies the constructive content of nonstandard theorems and corrects misconceptions about the axioms Transfer and Standard Part.

Findings

Some nonstandard theorems yield effective constructive results.

Certain axioms are not constructively valid as claimed by Wattenberg.

The paper refutes specific claims about Transfer and Standard Part axioms.

Abstract

Almost two decades ago, Wattenberg published a paper with the title 'Nonstandard Analysis and Constructivism?' in which he speculates on a possible connection between Nonstandard Analysis and constructive mathematics. We study Wattenberg's work in light of recent research on the aforementioned connection. On one hand, with only slight modification, some of Wattenberg's theorems in Nonstandard Analysis are seen to yield effective and constructive theorems (not involving Nonstandard Analysis). On the other hand, we establish the incorrectness of some of Wattenberg's (explicit and implicit) claims regarding the constructive status of the axioms Transfer and Standard Part of Nonstandard Analysis.