Set Theory and the Analyst

TL;DR

This survey explores the set-theoretic and logical foundations underlying the dualities and contrasts between Baire category and Lebesgue measure, providing essential background for analysts.

Contribution

It offers a comprehensive summary of set theory and logic relevant to understanding category-measure duality for practicing analysts.

Findings

Clarifies the set-theoretic basis of category and measure dualities

Highlights key logical principles relevant to analysis

Provides foundational knowledge for further research

Abstract

This survey is motivated by specific questions arising in the similarities and contrasts between (Baire) category and (Lebesgue) measure -- category-measure duality and non-duality, as it were. The bulk of the text is devoted to a summary, intended for the working analyst, of the extensive background in set theory and logic needed to discuss such matters: to quote from the Preface of Kelley [Kel]: "what every young analyst should know".