On a form of intrinsic optimism in Set Theory

TL;DR

This paper analyzes Gödel's phenomenological program for intrinsic optimism in set theory, distinguishing epistemological (phenomenological) analysis from mathematical deduction to advocate for new intrinsic axioms that may resolve CH. It develops a two-pronged account of naturalness (intrinsic and extrinsic) and defends objectivism while examining empirical-like criteria, along with Wang's examples and Hauser's related work. The discussion surveys skeptical positions (solved CH, vagueness, need for new primitives) and contrasts intrinsic optimism with extrinsic approaches (Woodin, Maddy), arguing for a flexible, open framework that accommodates multiple sources of justification. It also addresses practical concerns like sudden illumination and communicability, offering a nuanced stance that seeks to justify new axioms through phenomenological clarification and reflection principles while acknowledging challenges and alternative programs.

Abstract

In this paper we will try to provide a solid form of intrinsic set theoretical optimism. In other words, we will try to vindicate Gödel's views on phenomenology as a method for arriving at new axioms of ZFC in order to decide independent statements such as CH.