On some of Brouwer's axioms

TL;DR

This paper examines Brouwer's axioms and principles in intuitionistic mathematics, contrasting them with Bishop's constructive approach, and explores their implications in measure theory and combinatorics.

Contribution

It clarifies the role of Brouwer's principles in constructive mathematics and analyzes their consequences, highlighting differences from Bishop's methods.

Findings

Brouwer's principles lead to the Borel Hierarchy Theorem.

Brouwer's principles influence the development of the Intuitionistic Ramsey Theorem.

Differences between Brouwer's and Bishop's approaches in measure theory are discussed.

Abstract

We discuss the position of intuitionistic mathematics within the field of constructive mathematics. We discuss some principles defended and used by Brouwer but rejected by Bishop, like the Coninuity Principle, the Fan Theorem and the Bar Theorem. We explain some of their consequences in the development of constructive mathematics, like the Borel Hierarchy Theorem and the Intuitionistic Ramsey Theorem. We go into the theory of measure and integration as Bishop followed in this field a path different from Brouwer's.