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Second, the Proper Forcing Axiom implies that the class of uncountable linear orders has a five element basis. The elements are<jats:italic>X<\/jats:italic>, \u03c9<jats:sub>1<\/jats:sub>,<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1079898600003437_inline1\" \/>,<jats:italic>C<\/jats:italic>,<jats:italic>C *<\/jats:italic>where<jats:italic>X<\/jats:italic>is any suborder of the reals of size \u03c9<jats:sub>1<\/jats:sub>and<jats:italic>C<\/jats:italic>is any Countryman line. Third, the Proper Forcing Axiom implies the Singular Cardinals Hypothesis at<jats:italic>k<\/jats:italic>unless stationary subsets of<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1079898600003437_inline2\" \/>reflect. The techniques are expected to be applicable to other open problems concerning the theory of<jats:italic>H(\u03c9<jats:sub>2<\/jats:sub>)<\/jats:italic>.<\/jats:p>","DOI":"10.2178\/bsl\/1107959499","type":"journal-article","created":{"date-parts":[[2005,3,2]],"date-time":"2005-03-02T20:48:48Z","timestamp":1109796528000},"page":"51-60","source":"Crossref","is-referenced-by-count":1,"title":["Proper Forcing, Cardinal Arithmetic, and Uncountable Linear Orders"],"prefix":"10.1017","volume":"11","author":[{"given":"Justin Tatch","family":"Moore","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,1,15]]},"reference":[{"key":"S1079898600003437_ref016","unstructured":"Shelah Saharon , Reflection implies SCH, Preprint, 07 2004."},{"key":"S1079898600003437_ref025","volume-title":"Technical Report 13","author":"Todor\u010devi\u0107","year":"2000"},{"key":"S1079898600003437_ref027","unstructured":"Todorcevic Stevo , Coherent sequences, North-Holland, (in preparation)."},{"key":"S1079898600003437_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BF02761119"},{"key":"S1079898600003437_ref024","doi-asserted-by":"crossref","first-page":"43","DOI":"10.4171\/dms\/1-2\/4","article-title":"Basis problems in combinatorial set theory","volume":"II","author":"Todor\u010devi\u0107","year":"1998","journal-title":"Proceedings of the International Congress of Mathematicians, vol. 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