MartinsAxiom 2002-08-04 00:45:03 2003-02-04 17:58:38 Definition MA % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here %\PMlinkescapeword{theory} For any cardinal $\kappa$, \emph{Martin's Axiom for $\kappa$} ($MA_\kappa$) states that if $P$ is a partial order satisfying ccc then given any set of $\kappa$ \PMlinkname{dense}{DenseInAPoset} subsets of $P$, there is a \PMlinkname{directed}{DirectedSet} subset intersecting each such subset. \emph{Martin's Axiom} states that $MA_\kappa$ holds for every $\kappa<2^{\aleph_0}$.