G\"odel's Program in Set Theory

TL;DR

This paper explores G"odel's work on set theory, focusing on the quest for axioms like V=Ultimate-L to decide natural mathematical statements such as the Continuum Hypothesis, and discusses recent developments like the Sealing scenario.

Contribution

It provides an overview of G"odel's program in set theory, analyzing candidate axioms and their implications for resolving fundamental questions.

Findings

V=Ultimate-L as a promising axiom candidate

The Sealing scenario's recent impact on set theory

Open problems in axiomatic foundations for natural statements

Abstract

G\"odel proved in the 1930s in his famous Incompleteness Theorems that not all statements in mathematics can be proven or disproven from the accepted ZFC axioms. A few years later he showed the celebrated result that Cantor's Continuum Hypothesis is consistent. Afterwards, G\"odel raised the question whether, despite the fact that there is no reasonable axiomatic framework for all mathematical statements, natural statements, such as Cantor's Continuum Hypothesis, can be decided via extending ZFC by large cardinal axioms. While this question has been answered negatively, the problem of finding good axioms that decide natural mathematical statements remains open. There is a compelling candidate for an axiom that could solve G\"odel's problem: V = Ultimate-L. In addition, due to recent results the Sealing scenario has gained a lot of attention. We describe these candidates as well as their impact and relationship.