Forcing Axioms and Ω-logic

Abstract

It is well known that the Continuum Hypothesis is independent from the axioms of set theory by the results of Kurt Godel and Paul Cohen. However, there are some set theorists who do not consider that this independency of the hypothesis is the ultimate answer of the continuum problem and who think the validity of the hypothesis should be determined. Recently, several ideas have been proposed to decide the hypothesis. In this paper, we introduce two ideas to determine the truth of the Continuum Hypothesis, which are related to the concept of forcing absoluteness. One is the concepet of forcing axioms, and the other is Ω-logic.