Reflection principles, generic large cardinals, and the Continuum Problem

Abstract

Strong reflection principles with the reflection cardinal $\le {\aleph }_1$ 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.