Independence Phenomena in Mathematics: a Set Theoretic Perspective on Current Obstacles and Scenarios for Solutions

TL;DR

This paper discusses the limitations of ZFC set theory in resolving fundamental mathematical questions, reviews extensions and obstacles, and explores future directions for identifying suitable axioms.

Contribution

It provides an overview of current obstacles in set theory, recent progress, and potential solutions for understanding independence phenomena and selecting appropriate axioms.

Findings

Many set-theoretic problems remain undecidable within ZFC.

Extensions of ZFC are being studied to address these independence issues.

Progress has been made in understanding the impact of these extensions on the set-theoretic universe.

Abstract

The standard axioms of set theory, the Zermelo-Fraenkel axioms (ZFC), do not suffice to answer all questions in mathematics. While this follows abstractly from Kurt G\"odel's famous incompleteness theorems, we nowadays know numerous concrete examples for such questions. A large number of problems in set theory, for example, regularity properties such as Lebesgue measurability and the Baire property are not decided - for even rather simple (for example, projective) sets of reals - by ZFC. Even many problems outside of set theory have been showed to be unsolvable, meaning neither their truth nor their failure can be proven from ZFC. A major part of set theory is devoted to attacking this problem by studying various extensions of ZFC and their properties. We outline some of these extensions and explain current obstacles in understanding their impact on the set theoretical universe together with recent progress on these questions and future scenarios. This work is related to the overall goal to identify the "right" axioms for mathematics.