Martin's Axiom Versus the Continuum Hypothesis

Abstract

This chapter presents a comparison between Martin's axiom (MA) and the continuum hypothesis (CH). The statement and use of Martin's Axiom (MA) in mathematics is well known. MA is consistent with 2 א 0 > א l , and it is only in the presence of 2 א 0 > א l that MA becomes a powerful assumption. This method of obtaining independence results always leaves open the possibility that V = L can be weakened to CH. One way to resolve this problem is to formulate a MA type principle that has many of the consequences of MA + 2 א 0 > א l but is provably consistent with CH. The chapter reviews the progress made on the problem to date. Solovay and Tennenbaum proved that Souslin hypothesis (SA) could not be disproved by obtaining a model of ZFC in which there are no Souslin trees. The basic idea is that if a Souslin tree is forced with, then cardinals are preserved but ceases to be a Souslin tree. By iterating this forcing, one can eventually obtain a model in which there are no Souslin trees. Analyzing the Solovay–Tennenbaum proof, Martin, and independently Rowbottom and Kunen, were led to the formulation of MA.