MA has many different equivalent formulations and has been used very successfully to settle a large number of open problems in other areas of mathematics. For example, it implies Suslin’s Hypothesis and that every \(\mathbf{\Sigma}^1_2\) set of reals is Lebesgue measurable and has the Baire property. It also implies the negation of the CH and that \(2^{\aleph_0}\) is a regular cardinal, but it does not decide what cardinal it is. See Fremlin (1984) for many more consequences of MA and other equivalent formulations. In spite of this, the status of MA as an axiom of set theory is still unclear. Perhaps the most natural formulation of MA, from a foundational point of view, is in terms of reflection. Writing HC for the set of hereditarily-countable sets (i.e., countable sets whose elements are countable, the elements of which are also countable, and so on), MA is equivalent to: