Large cardinals and the continuum hypothesis
pp. 205-226
in: Carolin Antos, Radek Honzik, Claudio Ternullo, Sy D. Friedman (eds), The hyperuniverse project and maximality, Berlin, Springer, 2018Abstract
This is a survey paper which discusses the impact of large cardinals on provability of the Continuum Hypothesis (CH). It was Gödel who first suggested that perhaps "strong axioms of infinity" (large cardinals) could decide interesting set-theoretical statements independent over ZFC, such as CH. This hope proved largely unfounded for CH—one can show that virtually all large cardinals defined so far do not affect the status of CH. It seems to be an inherent feature of large cardinals that they do not determine properties of sets low in the cumulative hierarchy if such properties can be forced to hold or fail by small forcings.The paper can also be used as an introductory text on large cardinals as it defines all relevant concepts.