Reflection principles, generic large cardinals, and the Continuum Problem

TL;DR

This paper explores how strong reflection principles and generic large cardinals influence the possible sizes of the continuum, suggesting a trichotomy in continuum sizes under certain set-theoretic assumptions.

Contribution

It analyzes the implications of strong reflection principles and generic large cardinals on the continuum's size, proposing a trichotomy based on these principles.

Findings

Reflection principles with small reflection cardinals imply continuum size is $eth_1$, $eth_2$, or very large.

The study supports a trichotomy in continuum sizes under strong reflection assumptions.

Connections between reflection principles, generic large cardinals, and continuum size are established.

Abstract

Strong reflection principles with the reflection cardinal $\leq\aleph_1$ or $<2^{\aleph_0}$ imply that the size of the continuum is either $\aleph_1$ or $\aleph_2$ or very large. Thus, the stipulation, that a strong reflection principle should hold, seems to support the trichotomy on the possible size of the continuum. In this article, we examine the situation with the reflection principles and related notions of generic large cardinals.